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# On residual algebraic torsion extensions of a valuation of a field *K* to $K({x}_{1},\dots ,{x}_{n})$

- Figen Öke
^{1}Email author

**2013**:46

https://doi.org/10.1186/1687-1812-2013-46

© Öke; licensee Springer 2013

**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 $K({x}_{1},\dots ,{x}_{n})$ is described and its certain properties are investigated. Also, the existence of a residual algebraic torsion extension of a valuation on *K* to $K({x}_{1},\dots ,{x}_{n})$ 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 $K({x}_{1},\dots ,{x}_{n})$. 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 $K({x}_{1},\dots ,{x}_{n})$ 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 $K({x}_{1},\dots ,{x}_{n})$. In the third section, a residual algebraic torsion extension of *v* to $K({x}_{1},\dots ,{x}_{n})$ 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 $K({x}_{1},\dots ,{x}_{n})$ 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 ${k}_{v}$, $\overline{K}$ is an algebraic closure of *K*, $\overline{v}$ is a fixed extension of *v* to $\overline{K}$. The value group of $\overline{v}$ is the divisible closure of ${G}_{v}$ and its residue field is the algebraic closure of ${k}_{v}$. $K(x)$ and $K({x}_{1},\dots ,{x}_{n})$ are rational function fields over *K* with one and *n* variables respectively. For any *α* in ${O}_{v}$, ${\alpha}^{\ast}$ denotes its natural image in ${k}_{v}$. If ${a}_{1},\dots ,{a}_{n}\in \overline{K}$, then the restriction of $\overline{v}$ to $K({a}_{1},\dots ,{a}_{n})$ will be denoted by ${v}_{{a}_{1}\cdots {a}_{n}}$.

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 $F={\sum}_{t}{a}_{t}{x}^{t}\in K[x]$ as $w(F)={inf}_{t}(v({a}_{t}))$ is called Gauss extension of *v* to $K(x)$, its residue field is ${k}_{w}={k}_{v}({x}^{\ast})$, is the simple transcendental extension of ${k}_{v}$ and ${G}_{w}={G}_{v}$ [5].

is called a valuation defined by the pair $(a,\delta )\in \overline{K}\times {G}_{\overline{v}}$ or $(a,\delta )\in \overline{K}\times {G}_{\overline{v}}$ is called a pair of definitions of *w*. Also, *w* is an r.t. extension of *v*. If $[K(a):K]\le [K(b):K]$ for every $b\in \overline{K}$ such that $\overline{v}(b-a)\ge \delta $, then $(a,\delta )$ is called a minimal pair with respect to *K* [2].

*w*is an r.t. extension of

*v*to $K(x)$, there exists a minimal pair $(a,\delta )\in \overline{K}\times {G}_{\overline{v}}$ such that

*a*is separable over

*K*. Two pairs $({a}_{1},{\delta}_{1})$ and $({a}_{2},{\delta}_{2})$ define the same valuation

*w*if and only if ${\delta}_{1}={\delta}_{2}$ and $\overline{v}({a}_{1}-{a}_{2})\ge {\delta}_{1}$ [2]. Let $f=Irr(a,K)$ be the minimal polynomial of

*a*with respect to

*K*and $\gamma =w(f)$. For each $F\in K[x]$, let $F={F}_{1}+{F}_{2}f+\cdots +{F}_{n}{f}^{n}$, where ${F}_{t}\in K[x]$, $deg{F}_{t}<degf$, $t=1,\dots ,n$, be the

*f*-expansion of

*F*. Then

*w*is defined as follows:

Then ${G}_{w}={G}_{{v}_{a}}+Z\gamma $. Let *e* be the smallest non-zero positive integer such that $e\gamma \in {G}_{{v}_{a}}$. Then there exists $h\in K[x]$ such that $degh<degf$, ${v}_{a}(h(a))=e\gamma $ and $r={f}^{e}/h$ is an element of ${O}_{w}$ and ${r}^{\ast}\in {k}_{w}$ is transcendental over ${k}_{v}$. ${k}_{{v}_{a}}$ can be identified canonically with the algebraic closure of ${k}_{v}$ in ${k}_{w}$ and ${k}_{w}={k}_{{v}_{a}}({r}^{\ast})$ [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, ${G}_{v}\subseteq {G}_{w}\subseteq {G}_{\overline{v}}$ is satisfied [3].

The order relation on the set of all r.t. extensions of *v* to $K(x)$ is defined as follows: ${w}_{1}\le {w}_{2}\iff {w}_{1}(f)\le {w}_{2}(f)$ for all polynomials $f\in K[x]$. If ${w}_{1}\le {w}_{2}$ and there exists $f\in K[x]$ such that ${w}_{1}(f)<{w}_{2}(f)$, then it is written ${w}_{1}<{w}_{2}$. Let $({a}_{1},{\delta}_{1}),({a}_{2},{\delta}_{2})\in \overline{K}\times {G}_{\overline{v}}$ be minimal pairs of the definition of the r.t. extensions ${w}_{1}$ and ${w}_{2}$ of *v* to $K(x)$, respectively. Then ${w}_{1}\le {w}_{2}$ if and only if ${\delta}_{1}\le {\delta}_{2}$ and $\overline{v}({a}_{1}-{a}_{2})\ge {\delta}_{1}$; moreover, ${w}_{1}<{w}_{2}$ if and only if ${\delta}_{1}\le {\delta}_{2}$ and $v({a}_{1}-{a}_{2})>{\delta}_{1}$ [3].

Let *I* be a well-ordered set without the last element and ${({w}_{i})}_{i\in I}$ be an ordered system of r.t. extensions of *v* to $K(x)$, where ${w}_{i}$ is defined by a minimal pair $({a}_{i},{\delta}_{i})\in \overline{K}\times {G}_{\overline{v}}$ for all $i\in I$. If ${w}_{i}\le {w}_{j}$ for all $i<j$, then ${({w}_{i})}_{i\in I}$ is called an ordered system of r.t. extensions of *v* to $K(x)$.

for all $f\in K[x]$ is an extension of *v* to $K(x)$ and it is called a limit of the ordered system ${({w}_{i})}_{i\in I}$. *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 $K({x}_{1},\dots ,{x}_{n})$ can be defined. For this reason the r.t. extension of *v* to $K({x}_{1},\dots ,{x}_{n})$ defined in [4] can be used. An r.t. extension of *v* to $K({x}_{1},\dots ,{x}_{n})$ is defined by using r.t. extensions of *v* to $K({x}_{m})$ for $m=1,\dots ,n$ in [4].

Let ${u}_{m}$ be an r.t. extension of *v* to $K({x}_{m})$ defined by a minimal pair $({a}_{m},{\delta}_{m})\in \overline{K}\times {G}_{\overline{v}}$ for $m=1,\dots ,n$, where $[K({a}_{1},\dots ,{a}_{n}):K]={\prod}_{m=1}^{n}[K({a}_{m}):K]$ and ${f}_{m}=Irr({a}_{m},K)$, ${\gamma}_{m}={u}_{m}({f}_{m})$ for $m=1,\dots ,n$. Each polynomial $F\in K[{x}_{1},\dots ,{x}_{n}]$ can be uniquely written as $F={\sum}_{{t}_{1},\dots ,{t}_{n}}{F}_{{t}_{1}\cdots {t}_{n}}{f}_{1}^{{t}_{1}}\cdots {f}_{n}^{{t}_{n}}$, where ${F}_{{t}_{1}\cdots {t}_{n}}\in K[{x}_{1},\dots ,{x}_{n}]$, ${deg}_{{x}_{m}}{F}_{{t}_{1}\cdots {t}_{n}}<deg{f}_{m}$ for $m=1,\dots ,n$.

*w*defined as

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

In the next section, an r.a.t extension of *v* to $K({x}_{1},\dots ,{x}_{n})$ will be defined by using that r.t. extension.

## 3 A residual algebraic torsion extension of *v* to $K({x}_{1},\dots ,{x}_{n})$

*v*to $K({x}_{m})$ defined by a minimal pair $({a}_{m},{\delta}_{m})\in \overline{K}\times {G}_{\overline{v}}$ for $m=1,\dots ,n$, where $[K({a}_{1},\dots ,{a}_{n}):K]={\prod}_{m=1}^{n}[K({a}_{m}):K]$ and let

*u*be the r.t. extension of

*v*to $K({x}_{1},\dots ,{x}_{n})$ defined as in (4). Let ${u}_{m}^{\prime}$ be an r.t. extension of

*v*to $K({x}_{m})$ defined by a minimal pair $({a}_{m}^{\prime},{\delta}_{m}^{\prime})\in \overline{K}\times {G}_{\overline{v}}$ for $m=1,\dots ,n$, where $[K({a}_{1}^{\prime},\dots ,{a}_{n}^{\prime}):K]={\prod}_{m=1}^{n}[K({a}_{m}^{\prime}):K]$ and let ${u}^{\prime}$ be the r.t. extension of

*v*to $K({x}_{1},\dots ,{x}_{n})$ defined as in (4). A relation between such kind of r.t. extensions of

*v*to $K({x}_{1},\dots ,{x}_{n})$ can be defined so that $u\le {u}^{\prime}$ if and only if ${u}_{m}\le {u}_{m}^{\prime}$ for $m=1,\dots ,n$. This is an order relation, and if $u\le {u}^{\prime}$, then for each polynomial $F\in K[{x}_{1},\dots ,{x}_{n}]$, $u(F)\le {u}^{\prime}(F)$ is satisfied. Because, for $F={\sum}_{{t}_{1},\dots ,{t}_{n}}{d}_{{t}_{1}\cdots {t}_{n}}{x}_{1}^{{t}_{1}}\cdots {x}_{n}^{{t}_{n}}\in K[{x}_{1},\dots ,{x}_{n}]$,

*I*be a well-ordered set without the last element and ${({w}_{i})}_{i\in I}$ be an ordered system of r.t. extensions of

*v*to $K({x}_{1},\dots ,{x}_{n})$, 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 $({a}_{{i}_{m}},{\delta}_{{i}_{m}})\in \overline{K}\times {G}_{\overline{v}}$, where $[K({a}_{{i}_{1}},\dots ,{a}_{{i}_{n}}):K]={\prod}_{m=1}^{n}[K({a}_{{i}_{m}}):K]$ for all $i\in I$. If ${w}_{i}\le {w}_{j}$ for all $i<j$, then ${({w}_{i})}_{i\in I}$ is an ordered system of r.t. extensions of

*v*to $K({x}_{1},\dots ,{x}_{n})$. Then the valuation

*w*of $K({x}_{1},\dots ,{x}_{n})$ defined as

for all $F\in K[{x}_{1},\dots ,{x}_{n}]$ is an extension of *v* to $K({x}_{1},\dots ,{x}_{n})$ and it is called a limit of the ordered system ${({w}_{i})}_{i\in I}$.

If ${w}_{m}$ is the restriction of *w* to $K({x}_{m})$ for $m=1,\dots ,n$, then ${w}_{m}$ is the limit of the ordered system ${({w}_{{i}_{m}})}_{i\in I}$ of r.t. extensions of *v* to $K({x}_{m})$. Also, *w* is the common extension of ${w}_{1},\dots ,{w}_{n}$ to $K({x}_{1},\dots ,{x}_{n})$. 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 $K({x}_{1},\dots ,{x}_{n})$.

If $w={sup}_{i}{w}_{i}$ is a residual algebraic torsion extension of *v* to $K({x}_{1},\dots ,{x}_{n})$, then ${G}_{v}\subseteq {G}_{w}\subseteq {G}_{\overline{v}}$ is satisfied. Some other properties of *w* are studied below.

Denote the extension of ${w}_{i}$ to $\overline{K}({x}_{1},\dots ,{x}_{n})$ by ${\overline{w}}_{i}$ and the extension of ${w}_{{i}_{m}}$ to $\overline{K}({x}_{m})$ by ${\overline{w}}_{{i}_{m}}$ for $m=1,\dots ,n$ and for all $i\in I$.

**Theorem 3.1**

*Let*${({\overline{w}}_{i})}_{i\in I}$

*be an ordered system of r*.

*t*.

*extensions of*$\overline{v}$

*to*$\overline{K}({x}_{1},\dots ,{x}_{n})$,

*where*${\overline{w}}_{i}$

*is defined as in*(4),

*i*.

*e*., ${w}_{i}$

*is the r*.

*t*.

*extension of*

*v*

*to*$\overline{K}({x}_{1},\dots ,{x}_{n})$

*which is the common extension of*${w}_{{i}_{m}}$

*for*$m=1,\dots ,n$

*and for all*$i\in I$.

*Denote the restriction of*${\overline{w}}_{i}$

*to*$K({x}_{1},\dots ,{x}_{n})$

*by*${w}_{i}$

*and the restriction of*$\overline{v}$

*to*$K({a}_{{i}_{1}},\dots ,{a}_{{i}_{n}})$

*by*${v}_{i}={v}_{{a}_{{i}_{1}}\cdots {a}_{{i}_{n}}}$.

*Then*

- 1.
*For all*$i,j\in I$, $i<j$,*one has*${w}_{i}<{w}_{j}$,*i*.*e*., ${({w}_{i})}_{i\in I}$*is an ordered system of r*.*t*.*extensions of**v**to*$K({x}_{1},\dots ,{x}_{n})$. - 2.
*For all*$i,j\in I$, $i<j$,*one has*${k}_{{v}_{i}}\subseteq {k}_{{v}_{j}}$*and*${G}_{{v}_{i}}\subseteq {G}_{{v}_{j}}$. - 3.
*Suppose that*$\overline{w}={sup}_{i}{\overline{w}}_{i}$*and*$\overline{w}$*is not an r*.*t*.*extension of*$\overline{v}$*to*$\overline{K}({x}_{1},\dots ,{x}_{n})$*and denote that**w**is the restriction of*$\overline{w}$*to*$K({x}_{1},\dots ,{x}_{n})$.*Then*$w={sup}_{i}{w}_{i}$*and*${k}_{w}={\bigcup}_{i}{k}_{{v}_{i}}$*and*${G}_{w}={\bigcup}_{i}{G}_{{v}_{i}}$.

*Proof*For every $i\in I$ and $m=1,\dots ,n$, denote that ${f}_{{i}_{m}}=Irr({a}_{{i}_{m}},K)$ and ${deg}_{{x}_{m}}{f}_{{i}_{m}}={n}_{{i}_{m}}$.

- 1.
Since ${\overline{w}}_{i}<{\overline{w}}_{j}$ for all $i,j\in I$, $i<j$, we have ${w}_{i}\le {w}_{j}$. 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 $m=1,\dots ,n$, we have ${w}_{{i}_{m}}={w}_{{j}_{m}}$ for $m=1,\dots ,n$. Since $({a}_{{i}_{m}},{\delta}_{{i}_{m}})$ is a minimal pair of the definition of ${w}_{{i}_{m}}$, we have ${\delta}_{{i}_{m}}={\delta}_{{j}_{m}}$ for $m=1,\dots ,n$. But it is a contradiction, because ${({\overline{w}}_{{i}_{m}})}_{i\in I}$ is an ordered system of r.t. extensions of $\overline{v}$ to $\overline{K}({x}_{m})$ and so ${\overline{w}}_{{i}_{m}}<{\overline{w}}_{{j}_{m}}$,

*i.e.*, ${\delta}_{{i}_{m}}<{\delta}_{{j}_{m}}$ [3]. Hence ${w}_{{i}_{m}}<{w}_{{j}_{m}}$ for $m=1,\dots ,n$. Since ${w}_{i}$ and ${w}_{j}$ are common extensions of ${w}_{{i}_{m}}$ and ${w}_{{j}_{m}}$ respectively for $m=1,\dots ,n$ and for all $i\in I$, it is concluded that ${w}_{i}<{w}_{j}$ for all $i<j$. - 2.
It is enough to study for $B=F({a}_{{i}_{1}},\dots ,{a}_{{i}_{n}})\in K[{a}_{{i}_{1}},\dots ,{a}_{{i}_{n}}]$, where $F({x}_{1},\dots ,{x}_{n})\in K[{x}_{1},\dots ,{x}_{n}]$, ${deg}_{{x}_{m}}F({x}_{1},\dots ,{x}_{n})<{n}_{{i}_{m}}$.

It is seen that ${v}_{i}(B)={v}_{i}(F({a}_{{i}_{1}},\dots ,{a}_{{i}_{n}}))=\overline{v}(F({a}_{{i}_{1}},\dots ,{a}_{{i}_{n}}))={v}_{j}(F({a}_{{j}_{1}},\dots ,{a}_{{j}_{n}}))={v}_{j}(B)$ by using the [[3], Th. 2.3] and this gives ${G}_{{v}_{i}}\subseteq {G}_{{v}_{j}}$.

*B*in ${k}_{{v}_{j}}$. Hence ${k}_{{v}_{i}}\subseteq {k}_{{v}_{j}}$ for all $i,j\in I$, $i<j$.

- 3.
Since $\overline{w}={sup}_{i}{\overline{w}}_{i}$, we have $w={sup}_{i}{w}_{i}$ and

*w*is not an r.t. extension of*v*to $K({x}_{1},\dots ,{x}_{n})$. Using [[3], Th. 2.3] and the definition of ${w}_{{i}_{m}}$, the proof can be completed. Take $F({x}_{1},\dots ,{x}_{n})\in K[{x}_{1},\dots ,{x}_{n}]$ such that ${deg}_{{x}_{m}}F<{n}_{{i}_{m}}$. Since $({a}_{{i}_{m}},{\delta}_{{i}_{m}})$ is a minimal pair of the definition of ${w}_{{i}_{m}}$, $\overline{w}(F({x}_{1},\dots ,{x}_{n}))=w(F({x}_{1},\dots ,{x}_{n}))=\overline{v}(F({a}_{{i}_{1}},\dots ,{a}_{{i}_{n}}))={v}_{i}(F({a}_{{i}_{1}},\dots ,{a}_{{i}_{n}}))$. This means that ${G}_{{v}_{i}}\subseteq {G}_{w}$ for all $i\in I$ and so ${\bigcup}_{i}{G}_{{v}_{i}}\subseteq {G}_{w}$.

Conversely, let ${v}_{i}^{m}$ be the restriction of ${\overline{w}}_{i}$ to $K({a}_{{i}_{1}},\dots ,{a}_{{i}_{m-1}},{x}_{m},{a}_{{i}_{m+1}},\dots ,{a}_{{i}_{n}})$ for $m=1,\dots ,n$ and for all $i\in I$. Since ${v}_{i}^{m}(P({a}_{{i}_{1}},\dots ,{a}_{{i}_{m-1}},{x}_{m},{a}_{{i}_{m+1}},\dots ,{a}_{{i}_{n}}))={v}_{i}(P({a}_{{i}_{1}},\dots ,{a}_{{i}_{m}},\dots ,{a}_{{i}_{n}}))$, then $w(P({x}_{1},\dots ,{x}_{n}))={v}_{i}(P({a}_{{i}_{1}},\dots ,{a}_{{i}_{n}}))\in {G}_{{v}_{i}}$ for every $P({x}_{1},\dots ,{x}_{n})\in K[{x}_{1},\dots ,{x}_{n}]$. This gives ${G}_{w}\subseteq {\bigcup}_{i}{G}_{{v}_{i}}$.

Now, assume that $\overline{v}(F({a}_{{i}_{1}},\dots ,{a}_{{i}_{n}}))={v}_{i}(F({a}_{{i}_{1}},\dots ,{a}_{{i}_{m}}))=0$. Then $w(F({x}_{1},\dots ,{x}_{n}))=0$ and since ${deg}_{{x}_{m}}F({x}_{1},\dots ,{x}_{n})<{n}_{{i}_{m}}$, $F{({a}_{1},\dots ,{a}_{n})}^{\ast}$, which is ${v}_{i}$-residue of $F({x}_{1},\dots ,{x}_{n})$, coincides with the residue of $F({x}_{1},\dots ,{x}_{n})$ in ${k}_{w}$. This shows ${k}_{{v}_{i}}\subseteq {k}_{w}$ for all $i\in I$, and then ${\bigcup}_{i}{k}_{{v}_{i}}\subseteq {k}_{w}$ .

For the reverse inclusion, let $P({x}_{1},\dots ,{x}_{n})\in K[{x}_{1},\dots ,{x}_{n}]$ and $w(P({x}_{1},\dots ,{x}_{n}))=0$. For $m=1,\dots ,n$, $P{({a}_{{i}_{1}},\dots ,{a}_{{i}_{m-1}},{x}_{m},{a}_{{i}_{m+1}},\dots ,{a}_{{i}_{n}})}^{\ast}$ is equal to $P{({a}_{{i}_{1}},\dots ,{a}_{{i}_{m-1}},{a}_{{i}_{m}},{a}_{{i}_{m+1}},\dots ,{a}_{{i}_{n}})}^{\ast}\in {k}_{{v}_{i}}$ and so $P{({x}_{1},\dots ,{x}_{n})}^{\ast}=P{({a}_{{i}_{1}},\dots ,{a}_{{i}_{n}})}^{\ast}\in {k}_{{v}_{i}}$. Hence ${k}_{w}\subseteq {\bigcup}_{i}{k}_{{v}_{i}}$.

□

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*$K({x}_{1},\dots ,{x}_{n})$.

*Then the following are satisfied*:

- 1.
${G}_{{v}_{i}}\subseteq {G}_{{v}_{j}}$

*and*${k}_{{v}_{i}}\subseteq {k}_{{v}_{j}}$*for all*$i,j\in I$, $i<j$. - 2.
${({w}_{i})}_{i\in I}$

*is an ordered system of r*.*t*.*extensions of**v**to*$K({x}_{1},\dots ,{x}_{n})$*and*$w={sup}_{i}{w}_{i}$.*Moreover*,*we have*${k}_{w}={\bigcup}_{i}{k}_{{v}_{i}}$*and*${G}_{w}={\bigcup}_{i}{G}_{{v}_{i}}$.

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

## 4 Existence of r.a.t. extensions of valuations of *K* to $K({x}_{1},\dots ,{x}_{n})$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 $K({x}_{1},\dots ,{x}_{n})$, 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* ${G}_{v}\subset G$ *and* $G/{G}_{v}$ *is a countably infinite torsion group*. *Then there exists an r*.*a*.*t*. *extension* *w* *of* *v* *to* $K({x}_{1},\dots ,{x}_{n})$ *such that* ${k}_{w}\cong k$ *and* ${G}_{w}\cong G$.

*Proof* Since ${k}_{\overline{v}}$ is the algebraic closure of ${k}_{v}$, we have ${k}_{v}\subseteq k\subseteq {k}_{\overline{v}}$. Since $k/{k}_{v}$ is countably generated, there exists a tower of fields ${k}_{v}\subseteq {k}_{1}\subseteq {k}_{2}\subseteq \cdots $ such that ${\bigcup}_{s}{k}_{s}=k$, and since $G/{G}_{v}$ is a countable torsion group, there exists a sequence of subgroups of *G* such that ${G}_{v}={G}_{0}\subset {G}_{1}\subset {G}_{2}\subset \cdots \subset {G}_{s}\cdots \subset G$, ${G}_{s}\ne {G}_{s+1}$, ${G}_{s}/{G}_{v}$ is finite for all *s* and that ${\bigcup}_{s}{G}_{s}=G$. According to [[6], Th. 3.2], there exists an r.t. extension ${u}_{s}$ of *v* to $K({x}_{1},\dots ,{x}_{n})$ such that $transdeg{k}_{{u}_{s}}/{k}_{v}=n$, the algebraic closure of ${k}_{v}$ in ${k}_{{u}_{s}}$ is ${k}_{s}$, ${G}_{{u}_{s}}={G}_{s}$ and if $m\ne {m}^{\prime}$, then the restriction of ${u}_{s}$ to $K({x}_{m},{x}_{{m}^{\prime}})$ is not the Gauss extension of the restriction of ${u}_{s}$ to $K({x}_{m})$ for $m,{m}^{\prime}=1,\dots ,n$ and for all *s*. ${k}_{{u}_{s}}={k}_{s}({z}_{1},\dots ,{z}_{n})$, where ${z}_{m}$ is transcendental over ${k}_{s}$ for $m=1,\dots ,n$ 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 $m=1,\dots ,n$ and for all *s*. Then ${k}_{{u}_{{s}_{m}}}={k}_{{s}_{m}}({z}_{m})$, ${z}_{m}$ is transcendental over ${k}_{{s}_{m}}$ for $m=1,\dots ,n$ and for all *s*.

Then ${k}_{v}\subseteq {k}_{{1}_{m}}\subseteq {k}_{{2}_{m}}\subseteq \cdots \subseteq {k}_{{s}_{m}}\subseteq \cdots $ is the tower of finite extensions of ${k}_{v}$ for $m=1,\dots ,n$. Denote ${G}_{{u}_{{s}_{m}}}={G}_{{s}_{m}}$. ${G}_{v}\subset {G}_{{1}_{m}}\subset {G}_{{2}_{m}}\subset \cdots \subset {G}_{{s}_{m}}\subset \cdots \subset G$ is the sequence of subgroups of *G* such that ${G}_{{s}_{m}}\ne {G}_{{(s+1)}_{m}}$ and ${G}_{{s}_{m}}/{G}_{v}$ is finite for all *s* and for $m=1,\dots ,n$. Then there exists an r.a.t. extension ${w}_{m}$ of *v* to $K({x}_{m})$ such that ${k}_{{w}_{m}}\cong {\bigcup}_{s}{k}_{{s}_{m}}$ and ${G}_{{w}_{m}}\cong {\bigcup}_{s}{G}_{{s}_{m}}$ [3].

*K*, ${k}_{{w}_{1}}{k}_{{w}_{2}}/{k}_{{w}_{1}}$ is a countable generated infinite algebraic extension and $\u3008{G}_{{w}_{1}}\cup {G}_{{w}_{2}}\u3009/{G}_{{w}_{1}}$ 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 ${k}_{{v}_{2}}\cong {k}_{{w}_{1}}{k}_{{w}_{2}}$ and ${G}_{{v}_{2}}\cong \u3008{G}_{{w}_{1}}\cup {G}_{{w}_{2}}\u3009$. Using the induction on

*n*, it is obtained that there exits an r.a.t. extension ${v}_{n}=w$ of ${v}_{n-1}$ of $K({x}_{1},\dots ,{x}_{n-1})$ to $K({x}_{1},\dots ,{x}_{n})$ such that

Since ${v}_{i}$ is an r.a.t. extension of ${v}_{i-1}$ for $i=1,\dots ,n$, then ${v}_{n}=w$ is an r.a.t. extension of *v* to $K({x}_{1},\dots ,{x}_{n})$. □

**Theorem 4.2** *Let* $k/{k}_{v}$ *be a finite extension*, *G* *be an ordered group such that* ${G}_{v}\subset G$ *and* $G/{G}_{v}$ *is finite*. *Assume that* $trdeg\tilde{K}/K>0$. *Then there exists an r*.*a*.*t*. *extension of* *v* *to* $K({x}_{1},\dots ,{x}_{n})$ *such that* ${k}_{w}\cong k$ *and* ${G}_{w}\cong G$.

*Proof* Since $k/{k}_{v}$ is a finite extension, it can be written that $k={k}_{v}({b}_{1},\dots ,{b}_{t})$, where ${b}_{r}$ is algebraic over ${k}_{v}$ for $r=1,\dots ,t$. It can be taken $t\ge n$, 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 ${G}_{v}={H}_{0}\subset {H}_{1}\subset \cdots \subset {H}_{n}=G$ and ${H}_{r+1}/{H}_{r}$ is finite for $r=1,\dots ,n-1$.

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

Therefore if we denote the completion of $K({x}_{1})$ with respect to ${w}_{1}$ by $K({x}_{1}\tilde{)}$, then $trdegK({x}_{1}\tilde{)}/K({x}_{1})>0$. 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 ${k}_{{w}_{2}}\cong {k}_{v}({b}_{1},{b}_{2})$ and ${G}_{{w}_{2}}\cong {H}_{2}$. Using the induction, it is obtained that there exists an r.a.t. extension ${w}_{n-1}$ of ${w}_{n-2}$ on $K({x}_{1},\dots ,{x}_{n-2})$ to $K({x}_{1},\dots ,{x}_{n-1})$ such that its residue field is ${k}_{{w}_{n-1}}={k}_{v}({b}_{1},\dots ,{b}_{n-1})$ 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 $K({x}_{1},\dots ,{x}_{n})$ such that ${k}_{w}\cong {k}_{v}({b}_{n},\dots ,{b}_{t})=k$ and ${G}_{w}\cong G$. □

## Declarations

### Acknowledgements

Dedicated to Professor Hari M Srivastava.

## Authors’ Affiliations

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