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'henselian field'
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| Title of object: |
henselian field |
| Canonical Name: |
HenselianField |
| Type: |
Definition |
| Created on: |
2003-02-22 04:33:08 |
| Modified on: |
2007-06-02 10:08:49 |
| Classification: |
msc:11R99, msc:12J20, msc:13A18, msc:13F30 |
| Keywords: |
hensel, valuation, non archemidean |
| Defines: |
valuation ring, residue field, residue class field, Hensel property, henselian, henselisation |
Revision comment (for changes between this and next version):
| Changes for correction #12231 ('spelling'). |
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Content:
Let $\val$ be a non-archimedean valuation on a field $K$. Let
$V=\{x:|x|\le 1\}$. Since $\val$ is ultrametric, $V$ is closed under
addition and in fact an additive group. The other valuation axioms
ensure that $V$ is a ring. We call $V$ the \emph{valuation ring} of
$K$ with respect to the valuation $\val$. Note that the field of
fractions of $V$ is $K$.
The set $\mu=\{x:|x|<1\}$ is a maximal ideal of $V$. The factor
$R:=V/\mu$ is called the \emph{residue field} or the \emph{residue
class field}.
The map $\res:V \to V/\mu$ given by $x \mapsto x+\mu$ is called the
\emph{residue map}. We extend the definition of the residue map to
sequences of elements from $V$, and hence to $V[X]$ so that if $f(X)
\in V[X]$ is given by $\sum_{i \leq n} a_{i}X^{i}$ then $\res(f) \in
R[X]$ is given by $\sum_{i \leq n} \res(a{i})X^{i}$.
\bigskip
\par\noindent{\bf Hensel property:} Let $f(x) \in V[x]$. Suppose
$\res(f)(x)$ has a simple root $e \in k$. Then $f(x)$ has a root $e\PR
\in V$ and $\res(e\PR)=e$.
\medskip
Any valued field satisfying the Hensel property is called
\emph{henselian}. The completion of a non-archimidean valued field $K$
with respect to the valuation (cf. constructing the reals from the
rationals as the completion with respect to the standard metric) is a
henselian field.
Every non-archimedean valued field $K$ has a unique (up to
isomorphism) smallest henselian field $K^h$ containing it. We call
$K^h$ the \emph{henselisation} of $K$.
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