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A ramification filtration of the Galois group of a local field. III. (English. Russian original) Zbl 0918.11060

Izv. Math. 62, No. 5, 857-900 (1998); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 62, No. 5, 3-48 (1998).
This is an extension of previous papers on the topic by the author [V. Abrashkin, Tr. St.-Peterbg. Mat. Obshch. 3, 47-127 (1995; Zbl 0853.11096); Tr. Mat. Inst. Steklov 208, 18-69 (1995; Zbl 0884.11047)]. The problem solved in the paper is a description of the ramification filtration in the Galois group of an algebraic closure of a local field of finite characteristic \(p\). In this part the description is done modulo the closure of the subgroup of commutators of weight \(\geq p\). For this purpose, the author develops a nilpotent version of the Artin-Schreier theory. This gives an isomorphism between the quotient of the Galois group by the commutators of weight \(\geq p\) and a multiplicative group connected with the maximal quotient of nilpotency class \( < p\) of the free pro-finite Lie algebra over \(p\)-adic integers. Then the author defines an explicit chain of ideals in the Lie algebra and proves that it corresponds to the ramification filtration under the isomorphism. In the previous parts the theorem was established modulo \(p\)-powers and modulo commutators of weight \(\geq 3\).

MSC:

11S15 Ramification and extension theory
11S20 Galois theory
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