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On residual algebraic torsion extensions of a valuation of a field K to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq1_HTML.gif

Fixed Point Theory and Applications20132013:46

DOI: 10.1186/1687-1812-2013-46

Received: 11 December 2012

Accepted: 14 February 2013

Published: 5 March 2013

Abstract

Let v be a valuation of a field K with a value group G v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq2_HTML.gif and a residue field k v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq3_HTML.gif, w be an extension of v to K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq4_HTML.gif. Then w is called a residual algebraic torsion extension of v to K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq4_HTML.gif if k w / k v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq5_HTML.gif is an algebraic extension and G w / G v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq6_HTML.gif is a torsion group. In this paper, a residual algebraic torsion extension of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq1_HTML.gif is described and its certain properties are investigated. Also, the existence of a residual algebraic torsion extension of a valuation on K to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq1_HTML.gif with given residue field and value group is studied.

MSC:12J10, 12J20, 12F20.

Keywords

extensions of valuations residual algebraic torsion extensions valued fields value group residue field

1 Introduction

Let K be a field, v be a valuation on K with a value group G v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq2_HTML.gif and a residue field k v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq3_HTML.gif. The big target is to define all extensions of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq1_HTML.gif. Residual transcendental extensions of v to K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq4_HTML.gif are described by Popescu, Alexandru and Zaharescu in [1, 2]. Residual algebraic torsion extensions of v to K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq4_HTML.gif are studied for the first time in [3]. A residual transcendental extension of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq1_HTML.gif is defined in [4] by Öke. These studies are summarized in the second section. The paper is aimed to study residual algebraic torsion extensions of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq1_HTML.gif. In the third section, a residual algebraic torsion extension of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq1_HTML.gif is defined and certain properties of the residual algebraic torsion extensions given in [3] are generalized. In the last section, the existence of an r.a.t. extension of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq1_HTML.gif with given residue field and value group is demonstrated.

2 Preliminaries and some notations

Throughout this paper, v is a valuation of a field K with a value group G v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq2_HTML.gif, a valuation ring O v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq7_HTML.gif and a residue field k v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq8_HTML.gif, K ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq9_HTML.gif is an algebraic closure of K, v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq10_HTML.gif is a fixed extension of v to K ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq9_HTML.gif. The value group of v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq10_HTML.gif is the divisible closure of G v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq2_HTML.gif and its residue field is the algebraic closure of k v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq8_HTML.gif. K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq4_HTML.gif and K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq1_HTML.gif are rational function fields over K with one and n variables respectively. For any α in O v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq7_HTML.gif, α https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq11_HTML.gif denotes its natural image in k v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq3_HTML.gif. If a 1 , , a n K ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq12_HTML.gif, then the restriction of v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq10_HTML.gif to K ( a 1 , , a n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq13_HTML.gif will be denoted by v a 1 a n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq14_HTML.gif.

Let w be an extension of v to K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq4_HTML.gif. Then w is called a residual transcendental (r.t.) extension of v if k w / k v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq15_HTML.gif is a transcendental extension.

The valuation w, which is defined for each F = t a t x t K [ x ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq16_HTML.gif as w ( F ) = inf t ( v ( a t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq17_HTML.gif is called Gauss extension of v to K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq4_HTML.gif, its residue field is k w = k v ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq18_HTML.gif, is the simple transcendental extension of k v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq3_HTML.gif and G w = G v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq19_HTML.gif [5].

The valuation w ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq20_HTML.gif, which is defined for each F = t c t ( x a ) t K ¯ [ x ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq21_HTML.gif as
w ¯ ( F ) = inf t ( v ¯ ( c t ) + t δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_Equ1_HTML.gif
(1)

is called a valuation defined by the pair ( a , δ ) K ¯ × G v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq22_HTML.gif or ( a , δ ) K ¯ × G v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq23_HTML.gif is called a pair of definitions of w. Also, w is an r.t. extension of v. If [ K ( a ) : K ] [ K ( b ) : K ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq24_HTML.gif for every b K ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq25_HTML.gif such that v ¯ ( b a ) δ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq26_HTML.gif, then ( a , δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq27_HTML.gif is called a minimal pair with respect to K [2].

If w is an r.t. extension of v to K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq4_HTML.gif, there exists a minimal pair ( a , δ ) K ¯ × G v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq28_HTML.gif such that a is separable over K. Two pairs ( a 1 , δ 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq29_HTML.gif and ( a 2 , δ 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq30_HTML.gif define the same valuation w if and only if δ 1 = δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq31_HTML.gif and v ¯ ( a 1 a 2 ) δ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq32_HTML.gif [2]. Let f = Irr ( a , K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq33_HTML.gif be the minimal polynomial of a with respect to K and γ = w ( f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq34_HTML.gif. For each F K [ x ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq35_HTML.gif, let F = F 1 + F 2 f + + F n f n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq36_HTML.gif, where F t K [ x ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq37_HTML.gif, deg F t < deg f https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq38_HTML.gif, t = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq39_HTML.gif, be the f-expansion of F. Then w is defined as follows:
w ( F ) = inf t ( v a ( F t ( a ) ) + t γ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_Equ2_HTML.gif
(2)

Then G w = G v a + Z γ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq40_HTML.gif. Let e be the smallest non-zero positive integer such that e γ G v a https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq41_HTML.gif. Then there exists h K [ x ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq42_HTML.gif such that deg h < deg f https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq43_HTML.gif, v a ( h ( a ) ) = e γ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq44_HTML.gif and r = f e / h https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq45_HTML.gif is an element of O w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq46_HTML.gif and r k w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq47_HTML.gif is transcendental over k v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq3_HTML.gif. k v a https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq48_HTML.gif can be identified canonically with the algebraic closure of k v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq3_HTML.gif in k w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq49_HTML.gif and k w = k v a ( r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq50_HTML.gif [2].

Let w be an extension of v to K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq4_HTML.gif. w is called a residual algebraic (r.a.) extension of v if k w / k v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq15_HTML.gif is an algebraic extension. If w is an r.a. extension of v to K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq4_HTML.gif and G w / G v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq51_HTML.gif is not a torsion group, then w is called a residual algebraic free (r.a.f.) extension of v. In this case, the quotient group G w / G v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq52_HTML.gif is a free abelian group. More precisely, G w / G v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq52_HTML.gif is isomorphic to Z [3].

w is called a residual algebraic torsion (r.a.t) extension of v if w is an r.a. extension of v and G w / G v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq53_HTML.gif is a torsion group. In this case, G v G w G v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq54_HTML.gif is satisfied [3].

The order relation on the set of all r.t. extensions of v to K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq4_HTML.gif is defined as follows: w 1 w 2 w 1 ( f ) w 2 ( f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq55_HTML.gif for all polynomials f K [ x ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq56_HTML.gif. If w 1 w 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq57_HTML.gif and there exists f K [ x ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq56_HTML.gif such that w 1 ( f ) < w 2 ( f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq58_HTML.gif, then it is written w 1 < w 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq59_HTML.gif. Let ( a 1 , δ 1 ) , ( a 2 , δ 2 ) K ¯ × G v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq60_HTML.gif be minimal pairs of the definition of the r.t. extensions w 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq61_HTML.gif and w 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq62_HTML.gif of v to K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq4_HTML.gif, respectively. Then w 1 w 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq63_HTML.gif if and only if δ 1 δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq64_HTML.gif and v ¯ ( a 1 a 2 ) δ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq65_HTML.gif; moreover, w 1 < w 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq59_HTML.gif if and only if δ 1 δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq64_HTML.gif and v ( a 1 a 2 ) > δ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq66_HTML.gif [3].

Let I be a well-ordered set without the last element and ( w i ) i I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq67_HTML.gif be an ordered system of r.t. extensions of v to K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq4_HTML.gif, where w i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq68_HTML.gif is defined by a minimal pair ( a i , δ i ) K ¯ × G v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq69_HTML.gif for all i I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq70_HTML.gif. If w i w j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq71_HTML.gif for all i < j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq72_HTML.gif, then ( w i ) i I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq73_HTML.gif is called an ordered system of r.t. extensions of v to  K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq4_HTML.gif.

Then the valuation of K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq4_HTML.gif defined as
w ( f ) = sup i ( w i ( f ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_Equ3_HTML.gif
(3)

for all f K [ x ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq56_HTML.gif is an extension of v to K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq4_HTML.gif and it is called a limit of the ordered system ( w i ) i I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq73_HTML.gif. w may not be an r.t. extension of v to K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq4_HTML.gif [3].

Using the above studies an r.a.t extension of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq1_HTML.gif can be defined. For this reason the r.t. extension of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq1_HTML.gif defined in [4] can be used. An r.t. extension of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq1_HTML.gif is defined by using r.t. extensions of v to K ( x m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq74_HTML.gif for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq75_HTML.gif in [4].

Let u m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq76_HTML.gif be an r.t. extension of v to K ( x m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq77_HTML.gif defined by a minimal pair ( a m , δ m ) K ¯ × G v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq78_HTML.gif for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq79_HTML.gif, where [ K ( a 1 , , a n ) : K ] = m = 1 n [ K ( a m ) : K ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq80_HTML.gif and f m = Irr ( a m , K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq81_HTML.gif, γ m = u m ( f m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq82_HTML.gif for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq75_HTML.gif. Each polynomial F K [ x 1 , , x n ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq83_HTML.gif can be uniquely written as F = t 1 , , t n F t 1 t n f 1 t 1 f n t n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq84_HTML.gif, where F t 1 t n K [ x 1 , , x n ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq85_HTML.gif, deg x m F t 1 t n < deg f m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq86_HTML.gif for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq79_HTML.gif.

The valuation w defined as
u ( F ) = inf t 1 , , t n ( v a 1 a n ( F t 1 t 2 t n ( a 1 , , a n ) ) + t 1 γ 1 + + t n γ n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_Equ4_HTML.gif
(4)

is an extension of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq1_HTML.gif. u is an r.t. extension of v which is a common extension of u 1 , , u n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq87_HTML.gif to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq88_HTML.gif. Then G u = G v a 1 a n + Z γ 1 + + Z γ n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq89_HTML.gif. Let e m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq90_HTML.gif be the smallest positive integer such that e m γ m G v a m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq91_HTML.gif, where v a m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq92_HTML.gif is the restriction of v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq93_HTML.gif to K ( a m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq94_HTML.gif. Then there exists h m K [ x m ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq95_HTML.gif such that deg h m < deg f m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq96_HTML.gif, v a m ( h ( a m ) ) = e m γ m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq97_HTML.gif, r m = f m e m / h m O u m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq98_HTML.gif and r m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq99_HTML.gif is transcendental over k v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq8_HTML.gif for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq100_HTML.gif. k v a 1 a n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq101_HTML.gif can be canonically identified with the algebraic closure of k v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq8_HTML.gif in k w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq102_HTML.gif and k u = k v a 1 a n ( r 1 , , r n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq103_HTML.gif [4].

In the next section, an r.a.t extension of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq1_HTML.gif will be defined by using that r.t. extension.

3 A residual algebraic torsion extension of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq1_HTML.gif

Let u m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq76_HTML.gif be an r.t. extension of v to K ( x m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq77_HTML.gif defined by a minimal pair ( a m , δ m ) K ¯ × G v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq78_HTML.gif for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq75_HTML.gif, where [ K ( a 1 , , a n ) : K ] = m = 1 n [ K ( a m ) : K ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq80_HTML.gif and let u be the r.t. extension of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq88_HTML.gif defined as in (4). Let u m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq104_HTML.gif be an r.t. extension of v to K ( x m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq77_HTML.gif defined by a minimal pair ( a m , δ m ) K ¯ × G v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq105_HTML.gif for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq75_HTML.gif, where [ K ( a 1 , , a n ) : K ] = m = 1 n [ K ( a m ) : K ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq106_HTML.gif and let u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq107_HTML.gif be the r.t. extension of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq108_HTML.gif defined as in (4). A relation between such kind of r.t. extensions of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq108_HTML.gif can be defined so that u u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq109_HTML.gif if and only if u m u m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq110_HTML.gif for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq75_HTML.gif. This is an order relation, and if u u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq109_HTML.gif, then for each polynomial F K [ x 1 , , x n ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq111_HTML.gif, u ( F ) u ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq112_HTML.gif is satisfied. Because, for F = t 1 , , t n d t 1 t n x 1 t 1 x n t n K [ x 1 , , x n ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq113_HTML.gif,
u ( F ) = inf t 1 , , t n ( v ( d t 1 t n ) + t 1 u 1 ( x 1 ) + + t n u n ( x n ) ) inf t 1 , , t n ( v ( d t 1 t n ) + t 1 u 1 ( x 1 ) + + t n u n ( x n ) ) = u ( F ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_Equa_HTML.gif
Now, let I be a well-ordered set without the last element and ( w i ) i I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq114_HTML.gif be an ordered system of r.t. extensions of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq1_HTML.gif, where w i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq68_HTML.gif is defined as in (4), i.e., w i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq115_HTML.gif is the common extension of w i m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq116_HTML.gif, where w i m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq116_HTML.gif is the r.t. extension of v to K ( x m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq74_HTML.gif defined by the minimal pair ( a i m , δ i m ) K ¯ × G v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq117_HTML.gif, where [ K ( a i 1 , , a i n ) : K ] = m = 1 n [ K ( a i m ) : K ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq118_HTML.gif for all i I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq70_HTML.gif. If w i w j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq71_HTML.gif for all i < j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq72_HTML.gif, then ( w i ) i I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq73_HTML.gif is an ordered system of r.t. extensions of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq1_HTML.gif. Then the valuation w of K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq1_HTML.gif defined as
w ( F ) = sup i ( w i ( F ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_Equ5_HTML.gif
(5)

for all F K [ x 1 , , x n ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq111_HTML.gif is an extension of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq1_HTML.gif and it is called a limit of the ordered system ( w i ) i I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq73_HTML.gif.

If w m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq119_HTML.gif is the restriction of w to K ( x m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq74_HTML.gif for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq75_HTML.gif, then w m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq119_HTML.gif is the limit of the ordered system ( w i m ) i I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq120_HTML.gif of r.t. extensions of v to K ( x m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq74_HTML.gif. Also, w is the common extension of w 1 , , w n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq121_HTML.gif to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq1_HTML.gif. Since w m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq119_HTML.gif may not be an r.t. extension of v to K ( x m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq74_HTML.gif, then w may not be an r.t. extension of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq1_HTML.gif.

If w = sup i w i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq122_HTML.gif is a residual algebraic torsion extension of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq1_HTML.gif, then G v G w G v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq123_HTML.gif is satisfied. Some other properties of w are studied below.

Denote the extension of w i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq115_HTML.gif to K ¯ ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq124_HTML.gif by w ¯ i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq125_HTML.gif and the extension of w i m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq126_HTML.gif to K ¯ ( x m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq127_HTML.gif by w ¯ i m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq128_HTML.gif for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq100_HTML.gif and for all i I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq70_HTML.gif.

Theorem 3.1 Let ( w ¯ i ) i I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq129_HTML.gif be an ordered system of r.t. extensions of v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq93_HTML.gif to K ¯ ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq124_HTML.gif, where w ¯ i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq125_HTML.gif is defined as in (4), i.e., w i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq68_HTML.gif is the r.t. extension of v to K ¯ ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq124_HTML.gif which is the common extension of w i m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq130_HTML.gif for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq75_HTML.gif and for all i I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq131_HTML.gif. Denote the restriction of w ¯ i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq132_HTML.gif to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq88_HTML.gif by w i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq115_HTML.gif and the restriction of v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq10_HTML.gif to K ( a i 1 , , a i n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq133_HTML.gif by v i = v a i 1 a i n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq134_HTML.gif. Then
  1. 1.

    For all i , j I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq135_HTML.gif, i < j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq72_HTML.gif, one has w i < w j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq136_HTML.gif, i.e., ( w i ) i I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq73_HTML.gif is an ordered system of r.t. extensions of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq108_HTML.gif.

     
  2. 2.

    For all i , j I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq137_HTML.gif, i < j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq72_HTML.gif, one has k v i k v j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq138_HTML.gif and G v i G v j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq139_HTML.gif.

     
  3. 3.

    Suppose that w ¯ = sup i w ¯ i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq140_HTML.gif and w ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq141_HTML.gif is not an r.t. extension of v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq10_HTML.gif to K ¯ ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq124_HTML.gif and denote that w is the restriction of w ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq141_HTML.gif to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq108_HTML.gif. Then w = sup i w i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq142_HTML.gif and k w = i k v i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq143_HTML.gif and G w = i G v i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq144_HTML.gif.

     
Proof For every i I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq70_HTML.gif and m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq100_HTML.gif, denote that f i m = Irr ( a i m , K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq145_HTML.gif and deg x m f i m = n i m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq146_HTML.gif.
  1. 1.

    Since w ¯ i < w ¯ j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq147_HTML.gif for all i , j I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq135_HTML.gif, i < j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq72_HTML.gif, we have w i w j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq148_HTML.gif. We show that w i < w j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq136_HTML.gif. Assume that w i = w j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq149_HTML.gif. Since w i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq115_HTML.gif is the common extension of w i m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq130_HTML.gif and w j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq150_HTML.gif is the common extension of w j m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq151_HTML.gif for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq75_HTML.gif, we have w i m = w j m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq152_HTML.gif for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq75_HTML.gif. Since ( a i m , δ i m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq153_HTML.gif is a minimal pair of the definition of w i m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq126_HTML.gif, we have δ i m = δ j m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq154_HTML.gif for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq75_HTML.gif. But it is a contradiction, because ( w ¯ i m ) i I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq155_HTML.gif is an ordered system of r.t. extensions of v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq93_HTML.gif to K ¯ ( x m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq156_HTML.gif and so w ¯ i m < w ¯ j m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq157_HTML.gif, i.e., δ i m < δ j m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq158_HTML.gif [3]. Hence w i m < w j m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq159_HTML.gif for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq75_HTML.gif. Since w i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq115_HTML.gif and w j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq150_HTML.gif are common extensions of w i m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq160_HTML.gif and w j m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq161_HTML.gif respectively for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq75_HTML.gif and for all i I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq70_HTML.gif, it is concluded that w i < w j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq136_HTML.gif for all i < j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq162_HTML.gif.

     
  2. 2.

    It is enough to study for B = F ( a i 1 , , a i n ) K [ a i 1 , , a i n ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq163_HTML.gif, where F ( x 1 , , x n ) K [ x 1 , , x n ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq164_HTML.gif, deg x m F ( x 1 , , x n ) < n i m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq165_HTML.gif.

     

It is seen that v i ( B ) = v i ( F ( a i 1 , , a i n ) ) = v ¯ ( F ( a i 1 , , a i n ) ) = v j ( F ( a j 1 , , a j n ) ) = v j ( B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq166_HTML.gif by using the [[3], Th. 2.3] and this gives G v i G v j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq167_HTML.gif.

Assume that v i ( B ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq168_HTML.gif. Then v j ( B ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq169_HTML.gif. Since ( a i m , δ i m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq153_HTML.gif is a minimal pair of the definition of w i m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq116_HTML.gif for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq170_HTML.gif, we have B = F ( a i 1 , , a i m 1 , x m , a i m + 1 , , a i n ) = F ( a i 1 , , a i m 1 , a i m , a i m + 1 , , a i n ) k v i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq171_HTML.gif coincides with the F ( a j 1 , , a j m 1 , a j m , a j m + 1 , , a j n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq172_HTML.gif which is the residue of B in k v j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq173_HTML.gif. Hence k v i k v j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq174_HTML.gif for all i , j I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq137_HTML.gif, i < j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq72_HTML.gif.
  1. 3.

    Since w ¯ = sup i w ¯ i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq175_HTML.gif, we have w = sup i w i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq176_HTML.gif and w is not an r.t. extension of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq88_HTML.gif. Using [[3], Th. 2.3] and the definition of w i m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq116_HTML.gif, the proof can be completed. Take F ( x 1 , , x n ) K [ x 1 , , x n ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq164_HTML.gif such that deg x m F < n i m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq177_HTML.gif. Since ( a i m , δ i m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq153_HTML.gif is a minimal pair of the definition of w i m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq130_HTML.gif, w ¯ ( F ( x 1 , , x n ) ) = w ( F ( x 1 , , x n ) ) = v ¯ ( F ( a i 1 , , a i n ) ) = v i ( F ( a i 1 , , a i n ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq178_HTML.gif. This means that G v i G w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq179_HTML.gif for all i I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq70_HTML.gif and so i G v i G w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq180_HTML.gif.

     

Conversely, let v i m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq181_HTML.gif be the restriction of w ¯ i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq125_HTML.gif to K ( a i 1 , , a i m 1 , x m , a i m + 1 , , a i n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq182_HTML.gif for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq100_HTML.gif and for all i I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq70_HTML.gif. Since v i m ( P ( a i 1 , , a i m 1 , x m , a i m + 1 , , a i n ) ) = v i ( P ( a i 1 , , a i m , , a i n ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq183_HTML.gif, then w ( P ( x 1 , , x n ) ) = v i ( P ( a i 1 , , a i n ) ) G v i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq184_HTML.gif for every P ( x 1 , , x n ) K [ x 1 , , x n ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq185_HTML.gif. This gives G w i G v i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq186_HTML.gif.

Now, assume that v ¯ ( F ( a i 1 , , a i n ) ) = v i ( F ( a i 1 , , a i m ) ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq187_HTML.gif. Then w ( F ( x 1 , , x n ) ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq188_HTML.gif and since deg x m F ( x 1 , , x n ) < n i m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq189_HTML.gif, F ( a 1 , , a n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq190_HTML.gif, which is v i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq191_HTML.gif-residue of F ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq192_HTML.gif, coincides with the residue of F ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq192_HTML.gif in k w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq102_HTML.gif. This shows k v i k w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq193_HTML.gif for all i I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq70_HTML.gif, and then i k v i k w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq194_HTML.gif .

For the reverse inclusion, let P ( x 1 , , x n ) K [ x 1 , , x n ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq195_HTML.gif and w ( P ( x 1 , , x n ) ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq196_HTML.gif. For m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq100_HTML.gif, P ( a i 1 , , a i m 1 , x m , a i m + 1 , , a i n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq197_HTML.gif is equal to P ( a i 1 , , a i m 1 , a i m , a i m + 1 , , a i n ) k v i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq198_HTML.gif and so P ( x 1 , , x n ) = P ( a i 1 , , a i n ) k v i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq199_HTML.gif. Hence k w i k v i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq200_HTML.gif.

 □

The following theorem can be obtained as a result of Theorem 3.1.

Corollary 3.2 Under the above notations, let w be an r.a.t. extension of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq108_HTML.gif. Then the following are satisfied:
  1. 1.

    G v i G v j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq139_HTML.gif and k v i k v j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq201_HTML.gif for all i , j I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq135_HTML.gif, i < j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq72_HTML.gif.

     
  2. 2.

    ( w i ) i I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq73_HTML.gif is an ordered system of r.t. extensions of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq108_HTML.gif and w = sup i w i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq202_HTML.gif. Moreover, we have k w = i k v i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq203_HTML.gif and G w = i G v i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq204_HTML.gif.

     

Proof If w is an r.a.t. extension of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq108_HTML.gif, then w ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq141_HTML.gif is an r.a.t. extension v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq93_HTML.gif to K ¯ ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq205_HTML.gif and so w ¯ m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq206_HTML.gif is an r.a.t. extension of v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq93_HTML.gif to K ¯ ( x m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq156_HTML.gif for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq75_HTML.gif. We can take { δ i m } i I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq207_HTML.gif for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq100_HTML.gif as co-final well-ordered subsets of M w ¯ m = { w ¯ ( x m a ) | a K ¯ } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq208_HTML.gif. I has no last element because w ¯ m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq206_HTML.gif is not an r.t. extension of v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq10_HTML.gif. For every i I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq70_HTML.gif, choose the element ( a i m , δ i m ) K ¯ × G v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq209_HTML.gif such that for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq100_HTML.gif, w ¯ ( x m a i m ) = δ i m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq210_HTML.gif and [ K ( a i m ) : K ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq211_HTML.gif is the smallest possible for δ i m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq212_HTML.gif. This means that if c m K ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq213_HTML.gif such that w ¯ ( x m c m ) = δ i m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq214_HTML.gif, then [ K ( c m ) : K ] [ K ( a i m ) : K ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq215_HTML.gif. Then ( a i m , δ i m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq153_HTML.gif is a minimal pair of the definition of w ¯ i m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq128_HTML.gif with respect to K for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq100_HTML.gif. According to [[3], Th. 4.1], w ¯ i m < w ¯ j m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq216_HTML.gif if i < j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq72_HTML.gif, which means that ( w ¯ i m ) i I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq217_HTML.gif is an ordered system of r.t. extensions of v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq93_HTML.gif to K ¯ ( x m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq218_HTML.gif for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq100_HTML.gif and ( w ¯ i m ) i I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq217_HTML.gif has a limit w ¯ m = sup i w ¯ i m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq219_HTML.gif which is an r.a.t extension of v to K ¯ ( x m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq218_HTML.gif. For all i I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq70_HTML.gif, take w ¯ i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq125_HTML.gif as the common extension of w ¯ i m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq128_HTML.gif to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq108_HTML.gif and w ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq141_HTML.gif as the common extension of w ¯ m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq220_HTML.gif to K ¯ ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq124_HTML.gif for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq100_HTML.gif. Denote the restriction of w ¯ i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq132_HTML.gif to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq221_HTML.gif by w i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq115_HTML.gif and denote the restriction of w ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq141_HTML.gif to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq108_HTML.gif by w. In the same way as that in the proof of Theorem 3.1, it is seen that w i < w j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq136_HTML.gif for i , j I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq222_HTML.gif, i < j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq72_HTML.gif and w i < w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq223_HTML.gif for all i I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq70_HTML.gif and w = sup i w i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq142_HTML.gif. Moreover, k w = i k v i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq224_HTML.gif and G w = i G v i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq144_HTML.gif are satisfied. □

4 Existence of r.a.t. extensions of valuations of K to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq108_HTML.gifwith given residue field and value group

It can be concluded from section three and from [3] that if w is an r.a.t. extension of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq108_HTML.gif, then k w / k v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq5_HTML.gif is a countable generated infinite algebraic extension and G w / G v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq6_HTML.gif is a countable infinite torsion group. In this section, the converse is studied.

Theorem 4.1 Let k / k v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq225_HTML.gif be a countably generated infinite algebraic extension and G be an ordered group such that G v G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq226_HTML.gif and G / G v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq227_HTML.gif is a countably infinite torsion group. Then there exists an r.a.t. extension w of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq108_HTML.gif such that k w k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq228_HTML.gif and G w G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq229_HTML.gif.

Proof Since k v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq230_HTML.gif is the algebraic closure of k v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq3_HTML.gif, we have k v k k v ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq231_HTML.gif. Since k / k v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq225_HTML.gif is countably generated, there exists a tower of fields k v k 1 k 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq232_HTML.gif such that s k s = k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq233_HTML.gif, and since G / G v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq227_HTML.gif is a countable torsion group, there exists a sequence of subgroups of G such that G v = G 0 G 1 G 2 G s G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq234_HTML.gif, G s G s + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq235_HTML.gif, G s / G v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq236_HTML.gif is finite for all s and that s G s = G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq237_HTML.gif. According to [[6], Th. 3.2], there exists an r.t. extension u s https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq238_HTML.gif of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq108_HTML.gif such that trans deg k u s / k v = n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq239_HTML.gif, the algebraic closure of k v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq3_HTML.gif in k u s https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq240_HTML.gif is k s https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq241_HTML.gif, G u s = G s https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq242_HTML.gif and if m m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq243_HTML.gif, then the restriction of u s https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq238_HTML.gif to K ( x m , x m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq244_HTML.gif is not the Gauss extension of the restriction of u s https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq238_HTML.gif to K ( x m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq77_HTML.gif for m , m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq245_HTML.gif and for all s. k u s = k s ( z 1 , , z n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq246_HTML.gif, where z m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq247_HTML.gif is transcendental over k s https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq241_HTML.gif for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq248_HTML.gif and for all s. Denote the restriction of u s https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq238_HTML.gif to K ( x m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq249_HTML.gif by u s m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq250_HTML.gif and the algebraic closure of k v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq3_HTML.gif in k u s m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq251_HTML.gif by k s m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq252_HTML.gif for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq100_HTML.gif and for all s. Then k u s m = k s m ( z m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq253_HTML.gif, z m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq247_HTML.gif is transcendental over k s m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq254_HTML.gif for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq170_HTML.gif and for all s.

Then k v k 1 m k 2 m k s m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq255_HTML.gif is the tower of finite extensions of k v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq3_HTML.gif for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq100_HTML.gif. Denote G u s m = G s m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq256_HTML.gif. G v G 1 m G 2 m G s m G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq257_HTML.gif is the sequence of subgroups of G such that G s m G ( s + 1 ) m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq258_HTML.gif and G s m / G v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq259_HTML.gif is finite for all s and for m = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq75_HTML.gif. Then there exists an r.a.t. extension w m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq119_HTML.gif of v to K ( x m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq74_HTML.gif such that k w m s k s m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq260_HTML.gif and G w m s G s m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq261_HTML.gif [3].

It means that w m = sup s ( u s m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq262_HTML.gif. Since x 1 , x 2 , , x n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq263_HTML.gif are algebraic independent over K, k w 1 k w 2 / k w 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq264_HTML.gif is a countable generated infinite algebraic extension and G w 1 G w 2 / G w 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq265_HTML.gif is a countable torsion group. Hence there exists an r.a.t. extension v 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq266_HTML.gif of w 1 = v 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq267_HTML.gif to K ( x 1 , x 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq268_HTML.gif such that k v 2 k w 1 k w 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq269_HTML.gif and G v 2 G w 1 G w 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq270_HTML.gif. Using the induction on n, it is obtained that there exits an r.a.t. extension v n = w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq271_HTML.gif of v n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq272_HTML.gif of K ( x 1 , , x n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq273_HTML.gif to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq108_HTML.gif such that
k w = k v n k w 1 k w n = k w 1 k w n = m = 1 n ( s k u s m ) = s k u s https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_Equb_HTML.gif
and
G w = G v n G w 1 G w n = m = 1 n ( s G u s m ) = s G u s . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_Equc_HTML.gif

Since v i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq191_HTML.gif is an r.a.t. extension of v i 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq274_HTML.gif for i = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq275_HTML.gif, then v n = w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq276_HTML.gif is an r.a.t. extension of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq108_HTML.gif. □

Theorem 4.2 Let k / k v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq277_HTML.gif be a finite extension, G be an ordered group such that G v G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq226_HTML.gif and G / G v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq278_HTML.gif is finite. Assume that tr deg K ˜ / K > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq279_HTML.gif. Then there exists an r.a.t. extension of v to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq108_HTML.gif such that k w k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq280_HTML.gif and G w G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq229_HTML.gif.

Proof Since k / k v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq277_HTML.gif is a finite extension, it can be written that k = k v ( b 1 , , b t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq281_HTML.gif, where b r https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq282_HTML.gif is algebraic over k v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq8_HTML.gif for r = 1 , , t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq283_HTML.gif. It can be taken t n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq284_HTML.gif, because if t < n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq285_HTML.gif, n t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq286_HTML.gif elements can be chosen as equal. Since G / G v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq227_HTML.gif is finite, there exists a sequence of subgroups of G such that G v = H 0 H 1 H n = G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq287_HTML.gif and H r + 1 / H r https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq288_HTML.gif is finite for r = 1 , , n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq289_HTML.gif.

Hence there exists an r.a.t. extension w 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq61_HTML.gif of v to K ( x 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq290_HTML.gif such that k w 1 k v ( b 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq291_HTML.gif and G w 1 H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq292_HTML.gif [3]. Let K ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq293_HTML.gif be the completion of K with respect to v and v ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq294_HTML.gif be the extension of v to K ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq293_HTML.gif. According to [[7], Prop. 1], the completion of K ( x 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq290_HTML.gif with respect to w 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq295_HTML.gif is isomorphic to a field belonging to F c ( Ω ˜ / K ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq296_HTML.gif, where Ω ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq297_HTML.gif is the completion of the algebraic closure Ω of K ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq293_HTML.gif with respect to the unique extension of v ˜ ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq298_HTML.gif to Ω and F c ( Ω ˜ / K ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq296_HTML.gif is the set of complete fields L such that K ˜ L Ω ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq299_HTML.gif. Moreover, since tr deg K ˜ / K > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq279_HTML.gif, there exists an element a ˜ K ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq300_HTML.gif which is transcendental over K. That is, there exists a Cauchy sequence { a i } i I K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq301_HTML.gif which converges to a ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq302_HTML.gif.

Therefore if we denote the completion of K ( x 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq290_HTML.gif with respect to w 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq295_HTML.gif by K ( x 1 ) ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq303_HTML.gif, then tr deg K ( x 1 ) ˜ / K ( x 1 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq304_HTML.gif. Also, H 2 / H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq305_HTML.gif is finite, then there exists an r.a.t. extension w 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq306_HTML.gif of w 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq295_HTML.gif to K ( x 1 , x 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq268_HTML.gif such that k w 2 k v ( b 1 , b 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq307_HTML.gif and G w 2 H 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq308_HTML.gif. Using the induction, it is obtained that there exists an r.a.t. extension w n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq309_HTML.gif of w n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq310_HTML.gif on K ( x 1 , , x n 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq311_HTML.gif to K ( x 1 , , x n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq312_HTML.gif such that its residue field is k w n 1 = k v ( b 1 , , b n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq313_HTML.gif and its value group is G w n 1 = H n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq314_HTML.gif. Finally, there exists an r.a.t. extension w = w n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq315_HTML.gif of w n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq309_HTML.gif to K ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq108_HTML.gif such that k w k v ( b n , , b t ) = k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq316_HTML.gif and G w G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_Article_385_IEq229_HTML.gif. □

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Trakya University

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