Henselianとは 意味・読み方・使い方

語源

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

形容詞

Henselian (not comparable)

1. (algebra, of a ring または 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, Springer, 3rd Edition, page 203,

     An analogous result holds for Henselian fields. Recall that a field ${\displaystyle K}$ is said to be Henselian with respect to a valuation ${\displaystyle v}$ if ${\displaystyle v}$ has a unique extension (also denoted ${\displaystyle v}$) to every algebraic extension of ${\displaystyle K}$. […] It follows that every algebraic extension of an^[sic] Henselian field ${\displaystyle K}$ is Henselian.

     […] Every valued field ${\displaystyle (K,v)}$ has a minimal separable algebraic extension ${\displaystyle (K_{v},v)}$ which is Henselian. The valued field is unique up to a ${\displaystyle K}$-isomorphism and is called the Henselian closure of ${\displaystyle (K,v)}$ [Ribenboim, p. 176].

   • 2017, Arno Fehm, Franziska Jahnke, Recent progress on definability of Henselian valuations, 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.

参考

• Henselization
• Hensel ring

Further reading

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