# Difference between revisions of "Henselian"

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Satisfying the [[Hensel lemma|Hensel lemma]]. | Satisfying the [[Hensel lemma|Hensel lemma]]. | ||

− | If | + | If $K$ is a [[Field|field]] equipped with a [[Valuation|valuation]] $v$ and if the Hensel lemma is true for the valuation ring of $v$, then $(K,v)$ is called a Henselian field and $v$ is called a Henselian valuation. See also [[Henselization of a valued field|Henselization of a valued field]]; [[Ramification theory of valued fields|Ramification theory of valued fields]]. |

− | If | + | If $K$ is complete in the topology induced by $v$, and if its value group (cf. also [[Valuation|Valuation]]) is an [[Archimedean group|Archimedean group]], then $v$ is Henselian. But contrary to completeness, being Henselian can be axiomatized in a first-order language of valued fields (by an infinite set of axioms expressing the validity of the [[Hensel lemma|Hensel lemma]]). Moreover, a complete field need not be Henselian if its value group is not Archimedean. Therefore, in general valuation theory and in the [[Model theory of valued fields|model theory of valued fields]], being Henselian has turned out to be more appropriate than completeness. |

Several equivalent conditions for a field to be Henselian, including an [[Implicit function|implicit function]] theorem, are stated in [[#References|[a1]]]; see also [[#References|[a4]]]. For the connection with other topological fields (cf. [[Topological field|Topological field]]) satisfying the implicit function theorem, see [[#References|[a3]]]. | Several equivalent conditions for a field to be Henselian, including an [[Implicit function|implicit function]] theorem, are stated in [[#References|[a1]]]; see also [[#References|[a4]]]. For the connection with other topological fields (cf. [[Topological field|Topological field]]) satisfying the implicit function theorem, see [[#References|[a3]]]. |

## Latest revision as of 19:33, 28 April 2014

Satisfying the Hensel lemma.

If $K$ is a field equipped with a valuation $v$ and if the Hensel lemma is true for the valuation ring of $v$, then $(K,v)$ is called a Henselian field and $v$ is called a Henselian valuation. See also Henselization of a valued field; Ramification theory of valued fields.

If $K$ is complete in the topology induced by $v$, and if its value group (cf. also Valuation) is an Archimedean group, then $v$ is Henselian. But contrary to completeness, being Henselian can be axiomatized in a first-order language of valued fields (by an infinite set of axioms expressing the validity of the Hensel lemma). Moreover, a complete field need not be Henselian if its value group is not Archimedean. Therefore, in general valuation theory and in the model theory of valued fields, being Henselian has turned out to be more appropriate than completeness.

Several equivalent conditions for a field to be Henselian, including an implicit function theorem, are stated in [a1]; see also [a4]. For the connection with other topological fields (cf. Topological field) satisfying the implicit function theorem, see [a3].

It was proved by F.K. Schmidt in 1933 that if a field admits two independent non-trivial Henselian valuations, then it must be separably algebraically closed. For refinements of this result, see [a5]. For a further generalization and an application, see [a2].

See also Hensel ring.

#### References

[a1] | F.-V. Kuhlmann, "Valuation theory of fields, abelian groups and modules" , Algebra, Logic and Applications , Gordon&Breach (to appear) |

[a2] | F. Pop, "On Grothendieck's conjecture of birational anabelian geometry" Ann. of Math. , 138 (1994) pp. 145–182 |

[a3] | A. Prestel, M. Ziegler, "Model theoretic methods in the theory of topological fields" J. Reine Angew. Math. , 299/300 (1978) pp. 318–341 |

[a4] | P. Ribenboim, "Equivalent forms of Hensel's lemma" Expo. Math. , 3 (1985) pp. 3–24 |

[a5] | S. Warner, "Topological fields" , Mathematics Studies , 157 , North-Holland (1989) |

**How to Cite This Entry:**

Henselian.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Henselian&oldid=31961