Henselian
English
editAlternative forms
editEtymology
editFrom Hensel (surname) + -ian; after German mathematician Kurt Hensel (1861–1941).
Adjective
editHenselian (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, 3rd edition, Springer, page 203:
- An analogous result holds for Henselian fields. Recall that a field is said to be Henselian with respect to a valuation if has a unique extension (also denoted ) to every algebraic extension of . […] It follows that every algebraic extension of an[sic] Henselian field is Henselian.
[…] Every valued field has a minimal separable algebraic extension which is Henselian. The valued field is unique up to a -isomorphism and is called the Henselian closure of [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.
- 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,
See also
editFurther reading
edit- Henselian ring on Wikipedia.Wikipedia
- Hensel's lemma on Wikipedia.Wikipedia