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On residual algebraic torsion extensions of a valuation of a field K to K({x}_{1},\dots ,{x}_{n})
Fixed Point Theory and Applications volume 2013, Article number: 46 (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.
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}{(xa)}^{t}\in \overline{K}[x] as
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}(ba)\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 fexpansion of F. Then w is defined as follows:
Then {G}_{w}={G}_{{v}_{a}}+Z\gamma. Let e be the smallest nonzero 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 wellordered 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
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
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}],
Now, let I be a wellordered 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}}.
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}_{m1}},{x}_{m},{a}_{{i}_{m+1}},\dots ,{a}_{{i}_{n}})}^{\ast}=F{({a}_{{i}_{1}},\dots ,{a}_{{i}_{m1}},{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}_{m1}},{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}_{m1}},{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}_{m1}},{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}_{m1}},{x}_{m},{a}_{{i}_{m+1}},\dots ,{a}_{{i}_{n}})}^{\ast} is equal to P{({a}_{{i}_{1}},\dots ,{a}_{{i}_{m1}},{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 cofinal wellordered 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 \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}_{n1} of K({x}_{1},\dots ,{x}_{n1}) to K({x}_{1},\dots ,{x}_{n}) such that
and
Since {v}_{i} is an r.a.t. extension of {v}_{i1} 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, nt 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 ,n1.
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}_{n1} of {w}_{n2} on K({x}_{1},\dots ,{x}_{n2}) to K({x}_{1},\dots ,{x}_{n1}) such that its residue field is {k}_{{w}_{n1}}={k}_{v}({b}_{1},\dots ,{b}_{n1}) and its value group is {G}_{{w}_{n1}}={H}_{n1}. Finally, there exists an r.a.t. extension w={w}_{n} of {w}_{n1} 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. □
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Acknowledgements
Dedicated to Professor Hari M Srivastava.
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Öke, F. On residual algebraic torsion extensions of a valuation of a field K to K({x}_{1},\dots ,{x}_{n}). Fixed Point Theory Appl 2013, 46 (2013). https://doi.org/10.1186/16871812201346
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DOI: https://doi.org/10.1186/16871812201346