Open Access

On residual algebraic torsion extensions of a valuation of a field K to K ( x 1 , , x n )

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 and a residue field k v , w be an extension of v to K ( x ) . Then w is called a residual algebraic torsion extension of v to K ( x ) if k w / k v is an algebraic extension and G w / G v is a torsion group. In this paper, a residual algebraic torsion extension of v to K ( x 1 , , x n ) 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 ) 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 and a residue field k v . The big target is to define all extensions of v to K ( x 1 , , x n ) . Residual transcendental extensions of v to K ( x ) are described by Popescu, Alexandru and Zaharescu in [1, 2]. Residual algebraic torsion extensions of v to K ( x ) are studied for the first time in [3]. A residual transcendental extension of v to K ( x 1 , , x n ) 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 ) . In the third section, a residual algebraic torsion extension of v to K ( x 1 , , x n ) 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 ) 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 , a valuation ring O v and a residue field k v , K ¯ is an algebraic closure of K, v ¯ is a fixed extension of v to K ¯ . The value group of v ¯ is the divisible closure of G v and its residue field is the algebraic closure of k v . K ( x ) and K ( x 1 , , x n ) are rational function fields over K with one and n variables respectively. For any α in O v , α denotes its natural image in k v . If a 1 , , a n K ¯ , then the restriction of v ¯ to K ( a 1 , , a n ) will be denoted by v a 1 a n .

Let w be an extension of v to K ( x ) . Then w is called a residual transcendental (r.t.) extension of v if k w / k v is a transcendental extension.

The valuation w, which is defined for each F = t a t x t K [ x ] as w ( F ) = inf t ( v ( a t ) ) is called Gauss extension of v to K ( x ) , its residue field is k w = k v ( x ) , is the simple transcendental extension of k v and G w = G v [5].

The valuation w ¯ , which is defined for each F = t c t ( x a ) t K ¯ [ x ] as
w ¯ ( F ) = inf t ( v ¯ ( c t ) + t δ )
(1)

is called a valuation defined by the pair ( a , δ ) K ¯ × G v ¯ or ( a , δ ) K ¯ × G v ¯ is called a pair of definitions of w. Also, w is an r.t. extension of v. If [ K ( a ) : K ] [ K ( b ) : K ] for every b K ¯ such that v ¯ ( b a ) δ , then ( a , δ ) is called a minimal pair with respect to K [2].

If w is an r.t. extension of v to K ( x ) , there exists a minimal pair ( a , δ ) K ¯ × G v ¯ such that a is separable over K. Two pairs ( a 1 , δ 1 ) and ( a 2 , δ 2 ) define the same valuation w if and only if δ 1 = δ 2 and v ¯ ( a 1 a 2 ) δ 1 [2]. Let f = Irr ( a , K ) be the minimal polynomial of a with respect to K and γ = w ( f ) . For each F K [ x ] , let F = F 1 + F 2 f + + F n f n , where F t K [ x ] , deg F t < deg f , t = 1 , , n , be the f-expansion of F. Then w is defined as follows:
w ( F ) = inf t ( v a ( F t ( a ) ) + t γ ) .
(2)

Then G w = G v a + Z γ . Let e be the smallest non-zero positive integer such that e γ G v a . Then there exists h K [ x ] such that deg h < deg f , v a ( h ( a ) ) = e γ and r = f e / h is an element of O w and r k w is transcendental over k v . k v a can be identified canonically with the algebraic closure of k v in k w and k w = k v a ( r ) [2].

Let w be an extension of v to K ( x ) . w is called a residual algebraic (r.a.) extension of v if k w / k v is an algebraic extension. If w is an r.a. extension of v to K ( x ) and G w / G v 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 is a free abelian group. More precisely, G w / G v 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 is a torsion group. In this case, G v G w G v ¯ is satisfied [3].

The order relation on the set of all r.t. extensions of v to K ( x ) is defined as follows: w 1 w 2 w 1 ( f ) w 2 ( f ) for all polynomials f K [ x ] . If w 1 w 2 and there exists f K [ x ] such that w 1 ( f ) < w 2 ( f ) , then it is written w 1 < w 2 . Let ( a 1 , δ 1 ) , ( a 2 , δ 2 ) K ¯ × G v ¯ be minimal pairs of the definition of the r.t. extensions w 1 and w 2 of v to K ( x ) , respectively. Then w 1 w 2 if and only if δ 1 δ 2 and v ¯ ( a 1 a 2 ) δ 1 ; moreover, w 1 < w 2 if and only if δ 1 δ 2 and v ( a 1 a 2 ) > δ 1 [3].

Let I be a well-ordered set without the last element and ( w i ) i I be an ordered system of r.t. extensions of v to K ( x ) , where w i is defined by a minimal pair ( a i , δ i ) K ¯ × G v ¯ for all i I . If w i w j for all i < j , then ( w i ) i I is called an ordered system of r.t. extensions of v to  K ( x ) .

Then the valuation of K ( x ) defined as
w ( f ) = sup i ( w i ( f ) )
(3)

for all f K [ x ] is an extension of v to K ( x ) and it is called a limit of the ordered system ( w i ) i I . w may not be an r.t. extension of v to K ( x ) [3].

Using the above studies an r.a.t extension of v to K ( x 1 , , x n ) can be defined. For this reason the r.t. extension of v to K ( x 1 , , x n ) defined in [4] can be used. An r.t. extension of v to K ( x 1 , , x n ) is defined by using r.t. extensions of v to K ( x m ) for m = 1 , , n in [4].

Let u m be an r.t. extension of v to K ( x m ) defined by a minimal pair ( a m , δ m ) K ¯ × G v ¯ for m = 1 , , n , where [ K ( a 1 , , a n ) : K ] = m = 1 n [ K ( a m ) : K ] and f m = Irr ( a m , K ) , γ m = u m ( f m ) for m = 1 , , n . Each polynomial F K [ x 1 , , x n ] can be uniquely written as F = t 1 , , t n F t 1 t n f 1 t 1 f n t n , where F t 1 t n K [ x 1 , , x n ] , deg x m F t 1 t n < deg f m for m = 1 , , n .

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 )
(4)

is an extension of v to K ( x 1 , , x n ) . u is an r.t. extension of v which is a common extension of u 1 , , u n to K ( x 1 , , x n ) . Then G u = G v a 1 a n + Z γ 1 + + Z γ n . Let e m be the smallest positive integer such that e m γ m G v a m , where v a m is the restriction of v ¯ to K ( a m ) . Then there exists h m K [ x m ] such that deg h m < deg f m , v a m ( h ( a m ) ) = e m γ m , r m = f m e m / h m O u m and r m is transcendental over k v for m = 1 , , n . k v a 1 a n can be canonically identified with the algebraic closure of k v in k w and k u = k v a 1 a n ( r 1 , , r n ) [4].

In the next section, an r.a.t extension of v to K ( x 1 , , x n ) will be defined by using that r.t. extension.

3 A residual algebraic torsion extension of v to K ( x 1 , , x n )

Let u m be an r.t. extension of v to K ( x m ) defined by a minimal pair ( a m , δ m ) K ¯ × G v ¯ for m = 1 , , n , where [ K ( a 1 , , a n ) : K ] = m = 1 n [ K ( a m ) : K ] and let u be the r.t. extension of v to K ( x 1 , , x n ) defined as in (4). Let u m be an r.t. extension of v to K ( x m ) defined by a minimal pair ( a m , δ m ) K ¯ × G v ¯ for m = 1 , , n , where [ K ( a 1 , , a n ) : K ] = m = 1 n [ K ( a m ) : K ] and let u be the r.t. extension of v to K ( x 1 , , x n ) defined as in (4). A relation between such kind of r.t. extensions of v to K ( x 1 , , x n ) can be defined so that u u if and only if u m u m for m = 1 , , n . This is an order relation, and if u u , then for each polynomial F K [ x 1 , , x n ] , u ( F ) u ( F ) 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 ] ,
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 ) .
Now, let I be a well-ordered set without the last element and ( w i ) i I be an ordered system of r.t. extensions of v to K ( x 1 , , x n ) , where w i is defined as in (4), i.e., w i is the common extension of w i m , where w i m is the r.t. extension of v to K ( x m ) defined by the minimal pair ( a i m , δ i m ) K ¯ × G v ¯ , where [ K ( a i 1 , , a i n ) : K ] = m = 1 n [ K ( a i m ) : K ] for all i I . If w i w j for all i < j , then ( w i ) i I is an ordered system of r.t. extensions of v to K ( x 1 , , x n ) . Then the valuation w of K ( x 1 , , x n ) defined as
w ( F ) = sup i ( w i ( F ) )
(5)

for all F K [ x 1 , , x n ] is an extension of v to K ( x 1 , , x n ) and it is called a limit of the ordered system ( w i ) i I .

If w m is the restriction of w to K ( x m ) for m = 1 , , n , then w m is the limit of the ordered system ( w i m ) i I of r.t. extensions of v to K ( x m ) . Also, w is the common extension of w 1 , , w n to K ( x 1 , , x n ) . Since w m may not be an r.t. extension of v to K ( x m ) , then w may not be an r.t. extension of v to K ( x 1 , , x n ) .

If w = sup i w i is a residual algebraic torsion extension of v to K ( x 1 , , x n ) , then G v G w G v ¯ is satisfied. Some other properties of w are studied below.

Denote the extension of w i to K ¯ ( x 1 , , x n ) by w ¯ i and the extension of w i m to K ¯ ( x m ) by w ¯ i m for m = 1 , , n and for all i I .

Theorem 3.1 Let ( w ¯ i ) i I be an ordered system of r.t. extensions of v ¯ to K ¯ ( x 1 , , x n ) , where w ¯ i is defined as in (4), i.e., w i is the r.t. extension of v to K ¯ ( x 1 , , x n ) which is the common extension of w i m for m = 1 , , n and for all i I . Denote the restriction of w ¯ i to K ( x 1 , , x n ) by w i and the restriction of v ¯ to K ( a i 1 , , a i n ) by v i = v a i 1 a i n . Then
  1. 1.

    For all i , j I , i < j , one has w i < w j , i.e., ( w i ) i I is an ordered system of r.t. extensions of v to K ( x 1 , , x n ) .

     
  2. 2.

    For all i , j I , i < j , one has k v i k v j and G v i G v j .

     
  3. 3.

    Suppose that w ¯ = sup i w ¯ i and w ¯ is not an r.t. extension of v ¯ to K ¯ ( x 1 , , x n ) and denote that w is the restriction of w ¯ to K ( x 1 , , x n ) . Then w = sup i w i and k w = i k v i and G w = i G v i .

     
Proof For every i I and m = 1 , , n , denote that f i m = Irr ( a i m , K ) and deg x m f i m = n i m .
  1. 1.

    Since w ¯ i < w ¯ j for all i , j I , i < j , we have w i w j . We show that w i < w j . Assume that w i = w j . Since w i is the common extension of w i m and w j is the common extension of w j m for m = 1 , , n , we have w i m = w j m for m = 1 , , n . Since ( a i m , δ i m ) is a minimal pair of the definition of w i m , we have δ i m = δ j m for m = 1 , , n . But it is a contradiction, because ( w ¯ i m ) i I is an ordered system of r.t. extensions of v ¯ to K ¯ ( x m ) and so w ¯ i m < w ¯ j m , i.e., δ i m < δ j m [3]. Hence w i m < w j m for m = 1 , , n . Since w i and w j are common extensions of w i m and w j m respectively for m = 1 , , n and for all i I , it is concluded that w i < w j for all i < j .

     
  2. 2.

    It is enough to study for B = F ( a i 1 , , a i n ) K [ a i 1 , , a i n ] , where F ( x 1 , , x n ) K [ x 1 , , x n ] , deg x m F ( x 1 , , x n ) < n i m .

     

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 ) by using the [[3], Th. 2.3] and this gives G v i G v j .

Assume that v i ( B ) = 0 . Then v j ( B ) = 0 . Since ( a i m , δ i m ) is a minimal pair of the definition of w i m for m = 1 , , n , 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 coincides with the F ( a j 1 , , a j m 1 , a j m , a j m + 1 , , a j n ) which is the residue of B in k v j . Hence k v i k v j for all i , j I , i < j .
  1. 3.

    Since w ¯ = sup i w ¯ i , we have w = sup i w i and w is not an r.t. extension of v to K ( x 1 , , x n ) . Using [[3], Th. 2.3] and the definition of w i m , the proof can be completed. Take F ( x 1 , , x n ) K [ x 1 , , x n ] such that deg x m F < n i m . Since ( a i m , δ i m ) is a minimal pair of the definition of w i m , 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 ) ) . This means that G v i G w for all i I and so i G v i G w .

     

Conversely, let v i m be the restriction of w ¯ i to K ( a i 1 , , a i m 1 , x m , a i m + 1 , , a i n ) for m = 1 , , n and for all i I . 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 ) ) , then w ( P ( x 1 , , x n ) ) = v i ( P ( a i 1 , , a i n ) ) G v i for every P ( x 1 , , x n ) K [ x 1 , , x n ] . This gives G w i G v i .

Now, assume that v ¯ ( F ( a i 1 , , a i n ) ) = v i ( F ( a i 1 , , a i m ) ) = 0 . Then w ( F ( x 1 , , x n ) ) = 0 and since deg x m F ( x 1 , , x n ) < n i m , F ( a 1 , , a n ) , which is v i -residue of F ( x 1 , , x n ) , coincides with the residue of F ( x 1 , , x n ) in k w . This shows k v i k w for all i I , and then i k v i k w .

For the reverse inclusion, let P ( x 1 , , x n ) K [ x 1 , , x n ] and w ( P ( x 1 , , x n ) ) = 0 . For m = 1 , , n , P ( a i 1 , , a i m 1 , x m , a i m + 1 , , a i n ) is equal to P ( a i 1 , , a i m 1 , a i m , a i m + 1 , , a i n ) k v i and so P ( x 1 , , x n ) = P ( a i 1 , , a i n ) k v i . Hence k w i k v i .

 □

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 ) . Then the following are satisfied:
  1. 1.

    G v i G v j and k v i k v j for all i , j I , i < j .

     
  2. 2.

    ( w i ) i I is an ordered system of r.t. extensions of v to K ( x 1 , , x n ) and w = sup i w i . Moreover, we have k w = i k v i and G w = i G v i .

     

Proof If w is an r.a.t. extension of v to K ( x 1 , , x n ) , then w ¯ is an r.a.t. extension v ¯ to K ¯ ( x 1 , , x n ) and so w ¯ m is an r.a.t. extension of v ¯ to K ¯ ( x m ) for m = 1 , , n . We can take { δ i m } i I for m = 1 , , n as co-final well-ordered subsets of M w ¯ m = { w ¯ ( x m a ) | a K ¯ } . I has no last element because w ¯ m is not an r.t. extension of v ¯ . For every i I , choose the element ( a i m , δ i m ) K ¯ × G v ¯ such that for m = 1 , , n , w ¯ ( x m a i m ) = δ i m and [ K ( a i m ) : K ] is the smallest possible for δ i m . This means that if c m K ¯ such that w ¯ ( x m c m ) = δ i m , then [ K ( c m ) : K ] [ K ( a i m ) : K ] . Then ( a i m , δ i m ) is a minimal pair of the definition of w ¯ i m with respect to K for m = 1 , , n . According to [[3], Th. 4.1], w ¯ i m < w ¯ j m if i < j , which means that ( w ¯ i m ) i I is an ordered system of r.t. extensions of v ¯ to K ¯ ( x m ) for m = 1 , , n and ( w ¯ i m ) i I has a limit w ¯ m = sup i w ¯ i m which is an r.a.t extension of v to K ¯ ( x m ) . For all i I , take w ¯ i as the common extension of w ¯ i m to K ( x 1 , , x n ) and w ¯ as the common extension of w ¯ m to K ¯ ( x 1 , , x n ) for m = 1 , , n . Denote the restriction of w ¯ i to K ( x 1 , , x n ) by w i and denote the restriction of w ¯ to K ( x 1 , , x n ) by w. In the same way as that in the proof of Theorem 3.1, it is seen that w i < w j for i , j I , i < j and w i < w for all i I and w = sup i w i . Moreover, k w = i k v i and G w = i G v i are satisfied. □

4 Existence of r.a.t. extensions of valuations of K to K ( x 1 , , x n ) 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 K ( x 1 , , x n ) , then k w / k v is a countable generated infinite algebraic extension and G w / G v is a countable infinite torsion group. In this section, the converse is studied.

Theorem 4.1 Let k / k v be a countably generated infinite algebraic extension and G be an ordered group such that G v G and G / G v is a countably infinite torsion group. Then there exists an r.a.t. extension w of v to K ( x 1 , , x n ) such that k w k and G w G .

Proof Since k v ¯ is the algebraic closure of k v , we have k v k k v ¯ . Since k / k v is countably generated, there exists a tower of fields k v k 1 k 2 such that s k s = k , and since G / G v 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 , G s G s + 1 , G s / G v is finite for all s and that s G s = G . According to [[6], Th. 3.2], there exists an r.t. extension u s of v to K ( x 1 , , x n ) such that trans deg k u s / k v = n , the algebraic closure of k v in k u s is k s , G u s = G s and if m m , then the restriction of u s to K ( x m , x m ) is not the Gauss extension of the restriction of u s to K ( x m ) for m , m = 1 , , n and for all s. k u s = k s ( z 1 , , z n ) , where z m is transcendental over k s for m = 1 , , n and for all s. Denote the restriction of u s to K ( x m ) by u s m and the algebraic closure of k v in k u s m by k s m for m = 1 , , n and for all s. Then k u s m = k s m ( z m ) , z m is transcendental over k s m for m = 1 , , n and for all s.

Then k v k 1 m k 2 m k s m is the tower of finite extensions of k v for m = 1 , , n . Denote G u s m = G s m . G v G 1 m G 2 m G s m G is the sequence of subgroups of G such that G s m G ( s + 1 ) m and G s m / G v is finite for all s and for m = 1 , , n . Then there exists an r.a.t. extension w m of v to K ( x m ) such that k w m s k s m and G w m s G s m [3].

It means that w m = sup s ( u s m ) . Since x 1 , x 2 , , x n are algebraic independent over K, k w 1 k w 2 / k w 1 is a countable generated infinite algebraic extension and G w 1 G w 2 / G w 1 is a countable torsion group. Hence there exists an r.a.t. extension v 2 of w 1 = v 1 to K ( x 1 , x 2 ) such that k v 2 k w 1 k w 2 and G v 2 G w 1 G w 2 . Using the induction on n, it is obtained that there exits an r.a.t. extension v n = w of v n 1 of K ( x 1 , , x n 1 ) to K ( x 1 , , x n ) 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
and
G w = G v n G w 1 G w n = m = 1 n ( s G u s m ) = s G u s .

Since v i is an r.a.t. extension of v i 1 for i = 1 , , n , then v n = w is an r.a.t. extension of v to K ( x 1 , , x n ) . □

Theorem 4.2 Let k / k v be a finite extension, G be an ordered group such that G v G and G / G v is finite. Assume that tr deg K ˜ / K > 0 . Then there exists an r.a.t. extension of v to K ( x 1 , , x n ) such that k w k and G w G .

Proof Since k / k v is a finite extension, it can be written that k = k v ( b 1 , , b t ) , where b r is algebraic over k v for r = 1 , , t . It can be taken t n , because if t < n , n t elements can be chosen as equal. Since G / G v is finite, there exists a sequence of subgroups of G such that G v = H 0 H 1 H n = G and H r + 1 / H r is finite for r = 1 , , n 1 .

Hence there exists an r.a.t. extension w 1 of v to K ( x 1 ) such that k w 1 k v ( b 1 ) and G w 1 H 1 [3]. Let K ˜ be the completion of K with respect to v and v ˜ be the extension of v to K ˜ . According to [[7], Prop. 1], the completion of K ( x 1 ) with respect to w 1 is isomorphic to a field belonging to F c ( Ω ˜ / K ˜ ) , where Ω ˜ is the completion of the algebraic closure Ω of K ˜ with respect to the unique extension of v ˜ ¯ to Ω and F c ( Ω ˜ / K ˜ ) is the set of complete fields L such that K ˜ L Ω ˜ . Moreover, since tr deg K ˜ / K > 0 , there exists an element a ˜ K ˜ which is transcendental over K. That is, there exists a Cauchy sequence { a i } i I K which converges to a ˜ .

Therefore if we denote the completion of K ( x 1 ) with respect to w 1 by K ( x 1 ) ˜ , then tr deg K ( x 1 ) ˜ / K ( x 1 ) > 0 . Also, H 2 / H 1 is finite, then there exists an r.a.t. extension w 2 of w 1 to K ( x 1 , x 2 ) such that k w 2 k v ( b 1 , b 2 ) and G w 2 H 2 . Using the induction, it is obtained that there exists an r.a.t. extension w n 1 of w n 2 on K ( x 1 , , x n 2 ) to K ( x 1 , , x n 1 ) such that its residue field is k w n 1 = k v ( b 1 , , b n 1 ) and its value group is G w n 1 = H n 1 . Finally, there exists an r.a.t. extension w = w n of w n 1 to K ( x 1 , , x n ) such that k w k v ( b n , , b t ) = k and G w G . □

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

Authors’ Affiliations

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

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