A General Hensel's Lemma

✍ Scribed by S. Priess-Crampe; P. Ribenboim

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 114 KB
• Volume: 232
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.2000.8401

✦ Synopsis

We prove a general form of Hensel's lemma, for several polynomials in several variables. It contains as a particular case the result of Greenberg. The main tool in the proof is the fixed point theorem for spherically complete ultrametric spaces. The classical Hensel's lemma-proved for p-adic integers-was extended by Krull [9] for arbitrary valuation domains.

Nagata [11,12] further extended the result for local noetherian rings A, with maximal ideal M, which are complete in the linear topology having a neighbourhood basis of 0 consisting of the powers of M.

Lafon [10] considered the more general situation of Henselian couples A L , where L is an ideal contained in the Jacobson radical of A; see also Greco [4] who studied the relationship between different formulations of the property embodied in Hensel's lemma.

All the above results concerned polynomials in one indeterminate. Greenberg [5] extended Hensel's lemma for r polynomials in n indeterminates (where r, n may be larger than 1 and r ≤ n), having coefficients in a complete discrete valued field.

Further related results may be seen in Bourbaki [1], Iversen [7], and Fisher [3].