Henselization of a valued field
A minimal algebraic extension of a valued field
(i.e., a field
equipped with a valuation
) such that the valuation ring of
satisfies the Hensel lemma. This holds if and only if
admits a unique extension to every algebraic extension field of
(cf. [a2]). Therefore, Henselizations can be characterized as the decomposition fields of the extensions
of
to the separable-algebraic closure
of
(see Ramification theory of valued fields). The minimality is expressed by the following universal property of Henselizations: they admit a unique embedding over
in every other Henselian extension field of
(cf. [a2]). In particular, the Henselization of
is unique up to a valuation-preserving isomorphism over
; thus, it makes sense to denote it by
(there are some other notations in the literature). The extension
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) |
Henselization of a valued field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Henselization_of_a_valued_field&oldid=18068