Abstract
Let v be a valuation of a field K with a value group and a residue field , w be an extension of v to . Then w is called a residual algebraic torsion extension of v to if is an algebraic extension and is a torsion group. In this paper, a residual algebraic torsion extension of v to is described and its certain properties are investigated. Also, the existence of a residual algebraic torsion extension of a valuation on K to 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 field1 Introduction
Let K be a field, v be a valuation on K with a value group and a residue field . The big target is to define all extensions of v to . Residual transcendental extensions of v to are described by Popescu, Alexandru and Zaharescu in [1,2]. Residual algebraic torsion extensions of v to are studied for the first time in [3]. A residual transcendental extension of v to 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 . In the third section, a residual algebraic torsion extension of v to 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 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 , a valuation ring and a residue field , is an algebraic closure of K, is a fixed extension of v to . The value group of is the divisible closure of and its residue field is the algebraic closure of . and are rational function fields over K with one and n variables respectively. For any α in , denotes its natural image in . If , then the restriction of to will be denoted by .
Let w be an extension of v to . Then w is called a residual transcendental (r.t.) extension of v if is a transcendental extension.
The valuation w, which is defined for each as is called Gauss extension of v to , its residue field is , is the simple transcendental extension of and [5].
The valuation , which is defined for each as
is called a valuation defined by the pair or is called a pair of definitions of w. Also, w is an r.t. extension of v. If for every such that , then is called a minimal pair with respect to K[2].
If w is an r.t. extension of v to , there exists a minimal pair such that a is separable over K. Two pairs and define the same valuation w if and only if and [2]. Let be the minimal polynomial of a with respect to K and . For each , let , where , , , be the fexpansion of F. Then w is defined as follows:
Then . Let e be the smallest nonzero positive integer such that . Then there exists such that , and is an element of and is transcendental over . can be identified canonically with the algebraic closure of in and [2].
Let w be an extension of v to . w is called a residual algebraic (r.a.) extension of v if is an algebraic extension. If w is an r.a. extension of v to and 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 is a free abelian group. More precisely, 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 is a torsion group. In this case, is satisfied [3].
The order relation on the set of all r.t. extensions of v to is defined as follows: for all polynomials . If and there exists such that , then it is written . Let be minimal pairs of the definition of the r.t. extensions and of v to , respectively. Then if and only if and ; moreover, if and only if and [3].
Let I be a wellordered set without the last element and be an ordered system of r.t. extensions of v to , where is defined by a minimal pair for all . If for all , then is called an ordered system of r.t. extensions of v to .
Then the valuation of defined as
for all is an extension of v to and it is called a limit of the ordered system . w may not be an r.t. extension of v to [3].
Using the above studies an r.a.t extension of v to can be defined. For this reason the r.t. extension of v to defined in [4] can be used. An r.t. extension of v to is defined by using r.t. extensions of v to for in [4].
Let be an r.t. extension of v to defined by a minimal pair for , where and , for . Each polynomial can be uniquely written as , where , for .
The valuation w defined as
is an extension of v to . u is an r.t. extension of v which is a common extension of to . Then . Let be the smallest positive integer such that , where is the restriction of to . Then there exists such that , , and is transcendental over for . can be canonically identified with the algebraic closure of in and [4].
In the next section, an r.a.t extension of v to will be defined by using that r.t. extension.
3 A residual algebraic torsion extension of v to
Let be an r.t. extension of v to defined by a minimal pair for , where and let u be the r.t. extension of v to defined as in (4). Let be an r.t. extension of v to defined by a minimal pair for , where and let be the r.t. extension of v to defined as in (4). A relation between such kind of r.t. extensions of v to can be defined so that if and only if for . This is an order relation, and if , then for each polynomial , is satisfied. Because, for ,
Now, let I be a wellordered set without the last element and be an ordered system of r.t. extensions of v to , where is defined as in (4), i.e., is the common extension of , where is the r.t. extension of v to defined by the minimal pair , where for all . If for all , then is an ordered system of r.t. extensions of v to . Then the valuation w of defined as
for all is an extension of v to and it is called a limit of the ordered system .
If is the restriction of w to for , then is the limit of the ordered system of r.t. extensions of v to . Also, w is the common extension of to . Since may not be an r.t. extension of v to , then w may not be an r.t. extension of v to .
If is a residual algebraic torsion extension of v to , then is satisfied. Some other properties of w are studied below.
Denote the extension of to by and the extension of to by for and for all .
Theorem 3.1Letbe an ordered system of r.t. extensions ofto, whereis defined as in (4), i.e., is the r.t. extension ofvtowhich is the common extension offorand for all. Denote the restriction oftobyand the restriction oftoby. Then
1. For all, , one has, i.e., is an ordered system of r.t. extensions ofvto.
3. Suppose thatandis not an r.t. extension oftoand denote thatwis the restriction ofto. Thenandand.
Proof For every and , denote that and .
1. Since for all , , we have . We show that . Assume that . Since is the common extension of and is the common extension of for , we have for . Since is a minimal pair of the definition of , we have for . But it is a contradiction, because is an ordered system of r.t. extensions of to and so , i.e., [3]. Hence for . Since and are common extensions of and respectively for and for all , it is concluded that for all .
2. It is enough to study for , where , .
It is seen that by using the [[3], Th. 2.3] and this gives .
Assume that . Then . Since is a minimal pair of the definition of for , we have coincides with the which is the residue of B in . Hence for all , .
3. Since , we have and w is not an r.t. extension of v to . Using [[3], Th. 2.3] and the definition of , the proof can be completed. Take such that . Since is a minimal pair of the definition of , . This means that for all and so .
Conversely, let be the restriction of to for and for all . Since , then for every . This gives .
Now, assume that . Then and since , , which is residue of , coincides with the residue of in . This shows for all , and then .
For the reverse inclusion, let and . For , is equal to and so . Hence .
□
The following theorem can be obtained as a result of Theorem 3.1.
Corollary 3.2Under the above notations, letwbe an r.a.t. extension ofvto. Then the following are satisfied:
2. is an ordered system of r.t. extensions ofvtoand. Moreover, we haveand.
Proof If w is an r.a.t. extension of v to , then is an r.a.t. extension to and so is an r.a.t. extension of to for . We can take for as cofinal wellordered subsets of . I has no last element because is not an r.t. extension of . For every , choose the element such that for , and is the smallest possible for . This means that if such that , then . Then is a minimal pair of the definition of with respect to K for . According to [[3], Th. 4.1], if , which means that is an ordered system of r.t. extensions of to for and has a limit which is an r.a.t extension of v to . For all , take as the common extension of to and as the common extension of to for . Denote the restriction of to by and denote the restriction of to by w. In the same way as that in the proof of Theorem 3.1, it is seen that for , and for all and . Moreover, and are satisfied. □
4 Existence of r.a.t. extensions of valuations of K to 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 , then is a countable generated infinite algebraic extension and is a countable infinite torsion group. In this section, the converse is studied.
Theorem 4.1Letbe a countably generated infinite algebraic extension andGbe an ordered group such thatandis a countably infinite torsion group. Then there exists an r.a.t. extensionwofvtosuch thatand.
Proof Since is the algebraic closure of , we have . Since is countably generated, there exists a tower of fields such that , and since is a countable torsion group, there exists a sequence of subgroups of G such that , , is finite for all s and that . According to [[6], Th. 3.2], there exists an r.t. extension of v to such that , the algebraic closure of in is , and if , then the restriction of to is not the Gauss extension of the restriction of to for and for all s. , where is transcendental over for and for all s. Denote the restriction of to by and the algebraic closure of in by for and for all s. Then , is transcendental over for and for all s.
Then is the tower of finite extensions of for . Denote . is the sequence of subgroups of G such that and is finite for all s and for . Then there exists an r.a.t. extension of v to such that and [3].
It means that . Since are algebraic independent over K, is a countable generated infinite algebraic extension and is a countable torsion group. Hence there exists an r.a.t. extension of to such that and . Using the induction on n, it is obtained that there exits an r.a.t. extension of of to such that
and
Since is an r.a.t. extension of for , then is an r.a.t. extension of v to . □
Theorem 4.2Letbe a finite extension, Gbe an ordered group such thatandis finite. Assume that. Then there exists an r.a.t. extension ofvtosuch thatand.
Proof Since is a finite extension, it can be written that , where is algebraic over for . It can be taken , because if , elements can be chosen as equal. Since is finite, there exists a sequence of subgroups of G such that and is finite for .
Hence there exists an r.a.t. extension of v to such that and [3]. Let be the completion of K with respect to v and be the extension of v to . According to [[7], Prop. 1], the completion of with respect to is isomorphic to a field belonging to , where is the completion of the algebraic closure Ω of with respect to the unique extension of to Ω and is the set of complete fields L such that . Moreover, since , there exists an element which is transcendental over K. That is, there exists a Cauchy sequence which converges to .
Therefore if we denote the completion of with respect to by , then . Also, is finite, then there exists an r.a.t. extension of to such that and . Using the induction, it is obtained that there exists an r.a.t. extension of on to such that its residue field is and its value group is . Finally, there exists an r.a.t. extension of to such that and . □
Competing interests
The authors declare that they have no competing interests.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
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