Henselian

alternative henselian

1. not comparablealgebraWhich satisfies the criteria for (some formulation of) Hensel's lemma.

   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.

   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

   2 more examples

   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.

   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#95;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].

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