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# henselian field

Let $|\!\cdot\!|$ be a non-archimedean valuation on a field $K$. Let
$V=\{x:|x|\leq 1\}$. Since $|\!\cdot\!|$ 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 *valuation ring* of
$K$ with respect to the valuation $|\!\cdot\!|$. 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 *residue field* or the *residue
class field*.

The map $\res:V\to V/\mu$ given by $x\mapsto x+\mu$ is called the
*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 $\sum_{{i\leq n}}\res(a{i})X^{{i}}$.

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 $e^{{\prime}}\in V$ and $\res(e^{{\prime}})=e$.

Any valued field satisfying the Hensel property is called
*henselian*. The completion of a non-archimedean 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 *henselisation* of $K$.

## Mathematics Subject Classification

13F30*no label found*13A18

*no label found*11R99

*no label found*12J20

*no label found*

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