View source for Henselization of a valued field

A minimal algebraic extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110170/h1101701.png" /> of a valued field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110170/h1101702.png" /> (i.e., a [[Field|field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110170/h1101703.png" /> equipped with a [[Valuation|valuation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110170/h1101704.png" />) such that the valuation ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110170/h1101705.png" /> satisfies the [[Hensel lemma|Hensel lemma]]. This holds if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110170/h1101706.png" /> admits a unique extension to every algebraic extension field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110170/h1101707.png" /> (cf. [[#References|[a2]]]). Therefore, Henselizations can be characterized as the decomposition fields of the extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110170/h1101708.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110170/h1101709.png" /> to the separable-algebraic closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110170/h11017010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110170/h11017011.png" /> (see [[Ramification theory of valued fields|Ramification theory of valued fields]]). The minimality is expressed by the following universal property of Henselizations: they admit a unique embedding over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110170/h11017012.png" /> in every other Henselian extension field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110170/h11017013.png" /> (cf. [[#References|[a2]]]). In particular, the Henselization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110170/h11017014.png" /> is unique up to a valuation-preserving isomorphism over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110170/h11017015.png" />; thus, it makes sense to denote it by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110170/h11017016.png" /> (there are some other notations in the literature). The extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110170/h11017017.png" /> is immediate (see also [[Valuation|valuation]]); for an elegant proof, see [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Ribenboim,   "Théorie des valuations" , Presses Univ. Montréal  (1964)</TD></TR></table>