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)