# Henselization of a valued field

A minimal algebraic extension $ ( L,w ) $
of a valued field $ ( K,v ) $(
i.e., a field $ K $
equipped with a valuation $ v $)
such that the valuation ring of $ w $
satisfies the Hensel lemma. This holds if and only if $ w $
admits a unique extension to every algebraic extension field of $ L $(
cf. [a2]). Therefore, Henselizations can be characterized as the decomposition fields of the extensions $ v ^ {s} $
of $ v $
to the separable-algebraic closure $ K ^ {s} $
of $ K $(
see Ramification theory of valued fields). The minimality is expressed by the following universal property of Henselizations: they admit a unique embedding over $ K $
in every other Henselian extension field of $ ( K,v ) $(
cf. [a2]). In particular, the Henselization of $ ( K,v ) $
is unique up to a valuation-preserving isomorphism over $ K $;
thus, it makes sense to denote it by $ ( K ^ {H} ,v ^ {H} ) $(
there are some other notations in the literature). The extension $ v ^ {H} \mid v $
is immediate (see also valuation); for an elegant proof, see [a1].

#### References

[a1] | J. Ax, "A metamathematical approach to some problems in number theory, Appendix" D.J. Lewis (ed.) , Proc. Symp. Pure Math. , 20 , Amer. Math. Soc. (1971) pp. 161–190 |

[a2] | P. Ribenboim, "Théorie des valuations" , Presses Univ. Montréal (1964) |

**How to Cite This Entry:**

Henselization of a valued field.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Henselization_of_a_valued_field&oldid=47212