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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]]].
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A minimal algebraic extension  $  ( L,w ) $
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of a valued field  $  ( K,v ) $(
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i.e., a [[Field|field]] $  K $
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equipped with a [[Valuation|valuation]] $  v $)  
 +
such that the valuation ring of $  w $
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satisfies the [[Hensel lemma|Hensel lemma]]. This holds if and only if $  w $
 +
admits a unique extension to every algebraic extension field of $  L $(
 +
cf. [[#References|[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|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. [[#References|[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|valuation]]); for an elegant proof, see [[#References|[a1]]].
  
 
====References====
 
====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>
 
<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>

Latest revision as of 22:10, 5 June 2020


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
This article was adapted from an original article by F.-V. Kuhlmann (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article