Henselian

English

Alternative forms

• henselian

Etymology

Hensel (surname) +‎ -ian; after German mathematician Kurt Hensel (1861–1941).

Adjective

Henselian (not comparable)

1. (algebra, of a ring or field) Which satisfies the criteria for (some formulation of) Hensel's lemma.
   • 1989, N. Bourbaki, Commutative Algebra: Chapters 1-7, [1985, N. Bourbaki, Éléments de Mathématique Algébre Commutative 1-4 et 5-7, Masson], Springer, page 256,

     A local ring satisfying conditions (H) and (C) is called Henselian. Every complete Hausdorff local ring is Henselian. If A is Henselian and B is a commutative A-algebra which is a local ring and a finitely generated A-module, then B is Henselian.

   • 2008, Michael D. Fried, Moshe Jarden, Field Arithmetic, 3rd edition, Springer, page 203:

     An analogous result holds for Henselian fields. Recall that a field $K$ is said to be Henselian with respect to a valuation $v$ if $v$ has a unique extension (also denoted $v$) to every algebraic extension of $K$. […] It follows that every algebraic extension of an^[sic] Henselian field $K$ is Henselian.
     […] Every valued field $(K,v)$ has a minimal separable algebraic extension $(K_{v},v)$ which is Henselian. The valued field is unique up to a $K$-isomorphism and is called the Henselian closure of $(K,v)$ [Ribenboim, p. 176].

   • 2017, Arno Fehm, Franziska Jahnke, “Recent progress on definability of Henselian valuations”, in Fabrizio Broglia, Françoise Delon, Max Dickman, Danielle Gondard-Cozette, Victoria Ann Powers, editors, Ordered Algebraic Structures and Related Topics: International Conference, American Mathematical Society, page 136:

     Although the study of the definability of henselian valuations has a long history starting with J. Robinson, most of the results in this area were proven during the last few years.

See also

• Henselization
• Hensel ring

Further reading

• Henselian ring on Wikipedia.Wikipedia
• Hensel's lemma on Wikipedia.Wikipedia

Anagrams

• Hanelines