Meaning of Henselian | Babel Free

Definitions

Which satisfies the criteria for (some formulation of) Hensel's lemma.

not-comparable

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].”

“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.”

CEFR level