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=47212