On residual algebraic torsion extensions of a valuation of a field K to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq2_HTML.gif and a residue field http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq3_HTML.gif , w be an extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq4_HTML.gif . Then w is called a residual algebraic torsion extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq4_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq5_HTML.gif is an algebraic extension and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq6_HTML.gif is a torsion group. In this paper, a residual algebraic torsion extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq2_HTML.gif and a residue field http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq3_HTML.gif . The big target is to define all extensions of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq1_HTML.gif . Residual transcendental extensions of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq4_HTML.gif are described by Popescu, Alexandru and Zaharescu in [1, 2]. Residual algebraic torsion extensions of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq4_HTML.gif are studied for the first time in [3]. A residual transcendental extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq1_HTML.gif . In the third section, a residual algebraic torsion extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq2_HTML.gif , a valuation ring http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq7_HTML.gif and a residue field http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq8_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq9_HTML.gif is an algebraic closure of K, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq10_HTML.gif is a fixed extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq9_HTML.gif . The value group of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq10_HTML.gif is the divisible closure of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq2_HTML.gif and its residue field is the algebraic closure of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq8_HTML.gif . http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq4_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq1_HTML.gif are rational function fields over K with one and n variables respectively. For any α in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq7_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq11_HTML.gif denotes its natural image in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq3_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq12_HTML.gif , then the restriction of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq10_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq13_HTML.gif will be denoted by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq14_HTML.gif .

Let w be an extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq4_HTML.gif . Then w is called a residual transcendental (r.t.) extension of v if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq15_HTML.gif is a transcendental extension.

The valuation w, which is defined for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq16_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq17_HTML.gif is called Gauss extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq4_HTML.gif , its residue field is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq18_HTML.gif , is the simple transcendental extension of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq3_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq19_HTML.gif [5].

The valuation http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq20_HTML.gif , which is defined for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq21_HTML.gif as
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_Equ1_HTML.gif
(1)

is called a valuation defined by the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq22_HTML.gif or http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq23_HTML.gif is called a pair of definitions of w. Also, w is an r.t. extension of v. If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq24_HTML.gif for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq25_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq26_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq27_HTML.gif is called a minimal pair with respect to K[2].

If w is an r.t. extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq4_HTML.gif , there exists a minimal pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq28_HTML.gif such that a is separable over K. Two pairs http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq29_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq30_HTML.gif define the same valuation w if and only if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq31_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq32_HTML.gif [2]. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq33_HTML.gif be the minimal polynomial of a with respect to K and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq34_HTML.gif . For each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq35_HTML.gif , let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq36_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq37_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq38_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq39_HTML.gif , be the f-expansion of F. Then w is defined as follows:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_Equ2_HTML.gif
(2)

Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq40_HTML.gif . Let e be the smallest non-zero positive integer such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq41_HTML.gif . Then there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq42_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq43_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq44_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq45_HTML.gif is an element of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq46_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq47_HTML.gif is transcendental over http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq3_HTML.gif . http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq48_HTML.gif can be identified canonically with the algebraic closure of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq3_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq49_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq50_HTML.gif [2].

Let w be an extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq4_HTML.gif . w is called a residual algebraic (r.a.) extension of v if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq15_HTML.gif is an algebraic extension. If w is an r.a. extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq4_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq52_HTML.gif is a free abelian group. More precisely, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq53_HTML.gif is a torsion group. In this case, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq54_HTML.gif is satisfied [3].

The order relation on the set of all r.t. extensions of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq4_HTML.gif is defined as follows: http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq55_HTML.gif for all polynomials http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq56_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq57_HTML.gif and there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq56_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq58_HTML.gif , then it is written http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq59_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq60_HTML.gif be minimal pairs of the definition of the r.t. extensions http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq61_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq62_HTML.gif of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq4_HTML.gif , respectively. Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq63_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq64_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq65_HTML.gif ; moreover, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq59_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq64_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq66_HTML.gif [3].

Let I be a well-ordered set without the last element and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq67_HTML.gif be an ordered system of r.t. extensions of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq4_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq68_HTML.gif is defined by a minimal pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq69_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq70_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq71_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq72_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq73_HTML.gif is called an ordered system of r.t. extensions of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq4_HTML.gif .

Then the valuation of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq4_HTML.gif defined as
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_Equ3_HTML.gif
(3)

for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq56_HTML.gif is an extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq4_HTML.gif and it is called a limit of the ordered system http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq73_HTML.gif . w may not be an r.t. extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq4_HTML.gif [3].

Using the above studies an r.a.t extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq1_HTML.gif can be defined. For this reason the r.t. extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq1_HTML.gif defined in [4] can be used. An r.t. extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq1_HTML.gif is defined by using r.t. extensions of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq74_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq75_HTML.gif in [4].

Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq76_HTML.gif be an r.t. extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq77_HTML.gif defined by a minimal pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq78_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq79_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq80_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq81_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq82_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq75_HTML.gif . Each polynomial http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq83_HTML.gif can be uniquely written as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq84_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq85_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq86_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq79_HTML.gif .

The valuation w defined as
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_Equ4_HTML.gif
(4)

is an extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq1_HTML.gif . u is an r.t. extension of v which is a common extension of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq87_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq88_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq89_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq90_HTML.gif be the smallest positive integer such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq91_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq92_HTML.gif is the restriction of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq93_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq94_HTML.gif . Then there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq95_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq96_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq97_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq98_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq99_HTML.gif is transcendental over http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq8_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq100_HTML.gif . http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq101_HTML.gif can be canonically identified with the algebraic closure of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq8_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq102_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq103_HTML.gif [4].

In the next section, an r.a.t extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq1_HTML.gif will be defined by using that r.t. extension.

3 A residual algebraic torsion extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq1_HTML.gif

Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq76_HTML.gif be an r.t. extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq77_HTML.gif defined by a minimal pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq78_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq75_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq80_HTML.gif and let u be the r.t. extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq88_HTML.gif defined as in (4). Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq104_HTML.gif be an r.t. extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq77_HTML.gif defined by a minimal pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq105_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq75_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq106_HTML.gif and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq107_HTML.gif be the r.t. extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq108_HTML.gif defined as in (4). A relation between such kind of r.t. extensions of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq108_HTML.gif can be defined so that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq109_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq110_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq75_HTML.gif . This is an order relation, and if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq109_HTML.gif , then for each polynomial http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq111_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq112_HTML.gif is satisfied. Because, for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq113_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_Equa_HTML.gif
Now, let I be a well-ordered set without the last element and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq114_HTML.gif be an ordered system of r.t. extensions of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq1_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq68_HTML.gif is defined as in (4), i.e., http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq115_HTML.gif is the common extension of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq116_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq116_HTML.gif is the r.t. extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq74_HTML.gif defined by the minimal pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq117_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq118_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq70_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq71_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq72_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq73_HTML.gif is an ordered system of r.t. extensions of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq1_HTML.gif . Then the valuation w of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq1_HTML.gif defined as
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_Equ5_HTML.gif
(5)

for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq111_HTML.gif is an extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq1_HTML.gif and it is called a limit of the ordered system http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq73_HTML.gif .

If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq119_HTML.gif is the restriction of w to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq74_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq75_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq119_HTML.gif is the limit of the ordered system http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq120_HTML.gif of r.t. extensions of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq74_HTML.gif . Also, w is the common extension of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq121_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq1_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq119_HTML.gif may not be an r.t. extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq74_HTML.gif , then w may not be an r.t. extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq1_HTML.gif .

If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq122_HTML.gif is a residual algebraic torsion extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq1_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq123_HTML.gif is satisfied. Some other properties of w are studied below.

Denote the extension of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq115_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq124_HTML.gif by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq125_HTML.gif and the extension of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq126_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq127_HTML.gif by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq128_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq100_HTML.gif and for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq70_HTML.gif .

Theorem 3.1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq129_HTML.gif be an ordered system of r.t. extensions of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq93_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq124_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq125_HTML.gif is defined as in (4), i.e., http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq68_HTML.gif is the r.t. extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq124_HTML.gif which is the common extension of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq130_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq75_HTML.gif and for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq131_HTML.gif . Denote the restriction of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq132_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq88_HTML.gif by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq115_HTML.gif and the restriction of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq10_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq133_HTML.gif by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq134_HTML.gif . Then
  1. 1.

    For all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq135_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq72_HTML.gif , one has http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq136_HTML.gif , i.e., http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq73_HTML.gif is an ordered system of r.t. extensions of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq108_HTML.gif .

     
  2. 2.

    For all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq137_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq72_HTML.gif , one has http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq138_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq139_HTML.gif .

     
  3. 3.

    Suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq140_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq141_HTML.gif is not an r.t. extension of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq10_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq124_HTML.gif and denote that w is the restriction of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq141_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq108_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq142_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq143_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq144_HTML.gif .

     
Proof For every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq70_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq100_HTML.gif , denote that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq145_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq146_HTML.gif .
  1. 1.

    Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq147_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq135_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq72_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq148_HTML.gif . We show that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq136_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq149_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq115_HTML.gif is the common extension of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq130_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq150_HTML.gif is the common extension of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq151_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq75_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq152_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq75_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq153_HTML.gif is a minimal pair of the definition of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq126_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq154_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq75_HTML.gif . But it is a contradiction, because http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq155_HTML.gif is an ordered system of r.t. extensions of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq93_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq156_HTML.gif and so http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq157_HTML.gif , i.e., http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq158_HTML.gif [3]. Hence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq159_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq75_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq115_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq150_HTML.gif are common extensions of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq160_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq161_HTML.gif respectively for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq75_HTML.gif and for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq70_HTML.gif , it is concluded that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq136_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq162_HTML.gif .

     
  2. 2.

    It is enough to study for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq163_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq164_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq165_HTML.gif .

    It is seen that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq166_HTML.gif by using the [[3], Th. 2.3] and this gives http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq167_HTML.gif .

    Assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq168_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq169_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq153_HTML.gif is a minimal pair of the definition of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq116_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq170_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq171_HTML.gif coincides with the http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq172_HTML.gif which is the residue of B in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq173_HTML.gif . Hence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq174_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq137_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq72_HTML.gif .

     
  3. 3.

    Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq175_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq176_HTML.gif and w is not an r.t. extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq88_HTML.gif . Using [[3], Th. 2.3] and the definition of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq116_HTML.gif , the proof can be completed. Take http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq164_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq177_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq153_HTML.gif is a minimal pair of the definition of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq130_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq178_HTML.gif . This means that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq179_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq70_HTML.gif and so http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq180_HTML.gif .

    Conversely, let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq181_HTML.gif be the restriction of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq125_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq182_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq100_HTML.gif and for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq70_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq183_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq184_HTML.gif for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq185_HTML.gif . This gives http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq186_HTML.gif .

    Now, assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq187_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq188_HTML.gif and since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq189_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq190_HTML.gif , which is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq191_HTML.gif -residue of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq192_HTML.gif , coincides with the residue of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq192_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq102_HTML.gif . This shows http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq193_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq70_HTML.gif , and then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq194_HTML.gif .

    For the reverse inclusion, let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq195_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq196_HTML.gif . For http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq100_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq197_HTML.gif is equal to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq198_HTML.gif and so http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq199_HTML.gif . Hence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq108_HTML.gif . Then the following are satisfied:
  1. 1.

    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq139_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq201_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq135_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq72_HTML.gif .

     
  2. 2.

    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq73_HTML.gif is an ordered system of r.t. extensions of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq108_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq202_HTML.gif . Moreover, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq203_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq204_HTML.gif .

     

Proof If w is an r.a.t. extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq108_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq141_HTML.gif is an r.a.t. extension http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq93_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq205_HTML.gif and so http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq206_HTML.gif is an r.a.t. extension of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq93_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq156_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq75_HTML.gif . We can take http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq207_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq100_HTML.gif as co-final well-ordered subsets of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq208_HTML.gif . I has no last element because http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq206_HTML.gif is not an r.t. extension of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq10_HTML.gif . For every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq70_HTML.gif , choose the element http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq209_HTML.gif such that for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq100_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq210_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq211_HTML.gif is the smallest possible for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq212_HTML.gif . This means that if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq213_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq214_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq215_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq153_HTML.gif is a minimal pair of the definition of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq128_HTML.gif with respect to K for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq100_HTML.gif . According to [[3], Th. 4.1], http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq216_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq72_HTML.gif , which means that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq217_HTML.gif is an ordered system of r.t. extensions of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq93_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq218_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq100_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq217_HTML.gif has a limit http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq219_HTML.gif which is an r.a.t extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq218_HTML.gif . For all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq70_HTML.gif , take http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq125_HTML.gif as the common extension of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq128_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq108_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq141_HTML.gif as the common extension of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq220_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq124_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq100_HTML.gif . Denote the restriction of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq132_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq221_HTML.gif by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq115_HTML.gif and denote the restriction of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq141_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq108_HTML.gif by w. In the same way as that in the proof of Theorem 3.1, it is seen that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq136_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq222_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq72_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq223_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq70_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq142_HTML.gif . Moreover, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq224_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq144_HTML.gif are satisfied. □

4 Existence of r.a.t. extensions of valuations of K to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq108_HTML.gif with 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq108_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq5_HTML.gif is a countable generated infinite algebraic extension and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq6_HTML.gif is a countable infinite torsion group. In this section, the converse is studied.

Theorem 4.1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq225_HTML.gif be a countably generated infinite algebraic extension and G be an ordered group such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq226_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq227_HTML.gif is a countably infinite torsion group. Then there exists an r.a.t. extension w of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq108_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq228_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq229_HTML.gif .

Proof Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq230_HTML.gif is the algebraic closure of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq3_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq231_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq225_HTML.gif is countably generated, there exists a tower of fields http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq232_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq233_HTML.gif , and since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq227_HTML.gif is a countable torsion group, there exists a sequence of subgroups of G such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq234_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq235_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq236_HTML.gif is finite for all s and that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq237_HTML.gif . According to [[6], Th. 3.2], there exists an r.t. extension http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq238_HTML.gif of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq108_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq239_HTML.gif , the algebraic closure of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq3_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq240_HTML.gif is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq241_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq242_HTML.gif and if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq243_HTML.gif , then the restriction of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq238_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq244_HTML.gif is not the Gauss extension of the restriction of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq238_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq77_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq245_HTML.gif and for all s. http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq246_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq247_HTML.gif is transcendental over http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq241_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq248_HTML.gif and for all s. Denote the restriction of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq238_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq249_HTML.gif by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq250_HTML.gif and the algebraic closure of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq3_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq251_HTML.gif by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq252_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq100_HTML.gif and for all s. Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq253_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq247_HTML.gif is transcendental over http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq254_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq170_HTML.gif and for all s.

Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq255_HTML.gif is the tower of finite extensions of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq3_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq100_HTML.gif . Denote http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq256_HTML.gif . http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq257_HTML.gif is the sequence of subgroups of G such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq258_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq259_HTML.gif is finite for all s and for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq75_HTML.gif . Then there exists an r.a.t. extension http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq119_HTML.gif of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq74_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq260_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq261_HTML.gif [3].

It means that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq262_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq263_HTML.gif are algebraic independent over K, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq264_HTML.gif is a countable generated infinite algebraic extension and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq265_HTML.gif is a countable torsion group. Hence there exists an r.a.t. extension http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq266_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq267_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq268_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq269_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq270_HTML.gif . Using the induction on n, it is obtained that there exits an r.a.t. extension http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq271_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq272_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq273_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq108_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_Equb_HTML.gif
and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_Equc_HTML.gif

Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq191_HTML.gif is an r.a.t. extension of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq274_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq275_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq276_HTML.gif is an r.a.t. extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq108_HTML.gif . □

Theorem 4.2 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq277_HTML.gif be a finite extension, G be an ordered group such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq226_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq278_HTML.gif is finite. Assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq279_HTML.gif . Then there exists an r.a.t. extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq108_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq280_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq229_HTML.gif .

Proof Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq277_HTML.gif is a finite extension, it can be written that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq281_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq282_HTML.gif is algebraic over http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq8_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq283_HTML.gif . It can be taken http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq284_HTML.gif , because if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq285_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq286_HTML.gif elements can be chosen as equal. Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq227_HTML.gif is finite, there exists a sequence of subgroups of G such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq287_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq288_HTML.gif is finite for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq289_HTML.gif .

Hence there exists an r.a.t. extension http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq61_HTML.gif of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq290_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq291_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq292_HTML.gif [3]. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq293_HTML.gif be the completion of K with respect to v and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq294_HTML.gif be the extension of v to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq293_HTML.gif . According to [[7], Prop. 1], the completion of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq290_HTML.gif with respect to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq295_HTML.gif is isomorphic to a field belonging to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq296_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq297_HTML.gif is the completion of the algebraic closure Ω of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq293_HTML.gif with respect to the unique extension of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq298_HTML.gif to Ω and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq296_HTML.gif is the set of complete fields L such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq299_HTML.gif . Moreover, since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq279_HTML.gif , there exists an element http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq300_HTML.gif which is transcendental over K. That is, there exists a Cauchy sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq301_HTML.gif which converges to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq302_HTML.gif .

Therefore if we denote the completion of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq290_HTML.gif with respect to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq295_HTML.gif by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq303_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq304_HTML.gif . Also, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq305_HTML.gif is finite, then there exists an r.a.t. extension http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq306_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq295_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq268_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq307_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq308_HTML.gif . Using the induction, it is obtained that there exists an r.a.t. extension http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq309_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq310_HTML.gif on http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq311_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq312_HTML.gif such that its residue field is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq313_HTML.gif and its value group is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq314_HTML.gif . Finally, there exists an r.a.t. extension http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq315_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq309_HTML.gif to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq108_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq316_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq229_HTML.gif . □

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

Authors’ Affiliations

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

References

  1. Alexandru V, Popescu N, Zaharescu A: A theorem of characterization of residual transcendental extensions of a valuation. J. Math. Kyoto Univ. 1988,24(8):579–592.MathSciNet
  2. Alexandru V, Popescu N, Zaharescu A: Minimal pairs of definition of a residual transcendental extension of a valuation. J. Math. Kyoto Univ. 1990,30(2):207–225.MathSciNetMATH
  3. Alexandru V, Popescu N, Zaharescu A: All valuations on http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-46/MediaObjects/13663_2012_385_IEq4_HTML.gif . J. Math. Kyoto Univ. 1990,30(2):281–296.MathSciNetMATH
  4. Öke F: A residual transcendental extension of a valuation on K to . Appl. Math. Comput. 2011, 218:956–958.MathSciNetView ArticleMATH
  5. Bourbaki N: Algèbre commutative. Hermann, Paris; 1964:375–471.MATH
  6. Öke F: On extensions of valuations with given residue field and value group. Mathematika 2009, 55:191–197.MathSciNetView ArticleMATH
  7. Iovita A, Zaharescu A: Completions of r.a.t. valued fields of rational functions. J. Number Theory 1995, 50:202–205.MathSciNetView ArticleMATH

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