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henselian field (Definition)

Let $ \vert\!\cdot\!\vert$ be a non-archimedean valuation on a field $ K$. Let $ V=\{x:\vert x\vert\le 1\}$. Since $ \vert\!\cdot\!\vert$ 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 $ \vert\!\cdot\!\vert$. Note that the field of fractions of $ V$ is $ K$.

The set $ \mu=\{x:\vert x\vert<1\}$ is a maximal ideal of $ V$. The factor $ R:=V/\mu$ is called the residue field or the residue class field.

The map $ {\mathrm{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 $ {\mathrm{res}}(f) \in R[X]$ is given by $ \sum_{i \leq n} {\mathrm{res}}(a{i})X^{i}$.


Hensel property: Let $ f(x) \in V[x]$. Suppose $ {\mathrm{res}}(f)(x)$ has a simple root $ e \in k$. Then $ f(x)$ has a root $ e^{\prime} \in V$ and $ {\mathrm{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$.



"henselian field" is owned by mps. [ full author list (2) | owner history (1) ]
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See Also: valuation, valuation domain is local

Also defines:  valuation ring, residue field, residue class field, Hensel property, henselian, henselisation
Keywords:  hensel, valuation, non archimedean
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Cross-references: isomorphism, standard metric, rationals, reals, completion, root, simple root, sequences, residue, map, factor, maximal ideal, field of fractions, ring, axioms, additive group, addition, closed under, ultrametric, field, valuation, non-archimedean
There are 21 references to this entry.

This is version 6 of henselian field, born on 2003-02-22, modified 2007-06-02.
Object id is 4047, canonical name is HenselianField.
Accessed 8068 times total.

Classification:
AMS MSC: 11R99 (Number theory :: Algebraic number theory: global fields :: Miscellaneous)
 12J20 (Field theory and polynomials :: Topological fields :: General valuation theory)
 13A18 (Commutative rings and algebras :: General commutative ring theory :: Valuations and their generalizations)
 13F30 (Commutative rings and algebras :: Arithmetic rings and other special rings :: Valuation rings)
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