Henselian

English

Alternative forms

henselian

Etymology

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

Adjective

Henselian (not comparable)

(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, 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.

Anagrams

Hanelines

单词	Henselian
释义	Henselian English Alternative forms henselian Etymology Hensel (surname) +‎ -ian; after German mathematician Kurt Hensel (1861–1941). Adjective Henselian (not comparable) (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, 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. See also Henselization Hensel ring Further reading Henselian ring on Wikipedia.Wikipedia Hensel's lemma on Wikipedia.Wikipedia Anagrams Hanelines
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Henselian

English

Alternative forms

Etymology

Adjective

See also

Further reading

Anagrams