On residual algebraic torsion extensions of a valuation of a field K to $K(x_1,\dots ,x_n)$

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.

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].

The valuation $\overline{w}$, which is defined for each $F={\sum }_t{c}_t{(x-a)}^t\in \overline{K}[x]$ as

$$\overline{w}(F)=\underset{t}{inf}(\overline{v}(c_t)+t\delta )$$

(1)

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].

If 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:

$$w(F)=\underset{t}{inf}(v_a(F_t(a))+t\gamma ).$$

(2)

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 $K(x)$.

Then the valuation of $K(x)$ defined as

$$w(f)=\underset{i}{sup}(w_i(f))$$

(3)

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 $m=1,\dots ,n$.

The valuation w defined as

$$u(F)=\underset{t_1,\dots ,t_n}{inf}(v_{a_1\cdots a_n}(F_{t_1{t}_2\cdots t_n}(a_1,\dots ,a_n))+t_1{\gamma }_1+\cdots +t_n{\gamma }_n)$$

(4)

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 $K(x_1,\dots ,x_n)$

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 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 $F={\sum }_{t_1,\dots ,t_n}{d}_{t_1\cdots t_n}{x}_1^{t_1}\cdots x_n^{t_n}\in K[x_1,\dots ,x_n]$,

$$\begin{array}{rcl}u(F)& =& \underset{t_1,\dots ,t_n}{inf}(v(d_{t_1\cdots t_n})+t_1{u}_1(x_1)+\cdots +t_n{u}_n(x_n))\\ \le & \underset{t_1,\dots ,t_n}{inf}(v(d_{t_1\cdots t_n})+t_1{u}_1^{\prime }(x_1)+\cdots +t_n{u}_n^{\prime }(x_n))=u^{\prime }(F).\end{array}$$

Now, 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 $K(x_1,\dots ,x_n)$, 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

$$w(F)=\underset{i}{sup}(w_i(F))$$

(5)

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 $G_{v_i}\subseteq G_{v_j}$.

Assume that $v_i(B)=0$. Then $v_j(B)=0$. Since $(a_{i_m},{\delta }_{i_m})$ is a minimal pair of the definition of $w_{i_m}$ for $m=1,\dots ,n$, we have $B^{\ast }=F{(a_{i_1},\dots ,a_{i_{m-1}},x_m,a_{i_{m+1}},\dots ,a_{i_n})}^{\ast }=F{(a_{i_1},\dots ,a_{i_{m-1}},a_{i_m},a_{i_{m+1}},\dots ,a_{i_n})}^{\ast }\in k_{v_i}$ coincides with the $F{(a_{j_1},\dots ,a_{j_{m-1}},a_{j_m},a_{j_{m+1}},\dots ,a_{j_n})}^{\ast }$ which is the residue of 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].

It means that $w_m={sup}_s(u_{s_m})$. Since $x_1,x_2,\dots ,x_n$ are algebraic independent over K, $k_{w_1}{k}_{w_2}/k_{w_1}$ is a countable generated infinite algebraic extension and $〈G_{w_1}\cup G_{w_2}〉/G_{w_1}$ 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 $k_{v_2}\cong k_{w_1}{k}_{w_2}$ and $G_{v_2}\cong 〈G_{w_1}\cup G_{w_2}〉$. Using the induction on n, it is obtained that there exits an r.a.t. extension $v_n=w$ of $v_{n-1}$ of $K(x_1,\dots ,x_{n-1})$ to $K(x_1,\dots ,x_n)$ such that

$$k_w=k_{v_n}\cong k_{w_1}\cdots k_{w_n}=〈k_{w_1}\cup \cdots \cup k_{w_n}〉=〈\bigcup _{m=1}^n\left(\bigcup _s{k}_{u_{s_m}}\right)〉=\bigcup _s{k}_{u_s}$$

and

$$G_w=G_{v_n}\cong 〈G_{w_1}\cup \cdots \cup G_{w_n}〉=〈\bigcup _{m=1}^n\left(\bigcup _s{G}_{u_{s_m}}\right)〉=\bigcup _s{G}_{u_s}.$$

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 $K(x_1\widetilde{)}$, then $trdegK(x_1\widetilde{)}/K(x_1)>0$. 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 $G_w\cong G$. □