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RAMIFICATION FILTRATION OF THE GALOIS GROUP OF A LOCAL FIELD. III
 

Summary: RAMIFICATION FILTRATION OF THE
GALOIS GROUP OF A LOCAL FIELD. III
Victor A. Abrashkin
Let K be a complete discrete valuation field of characteristic p > 0 with a finite
residue field and let #(p) be the Galois group of its maximal p­extension. The main
result of the paper describes the image of the ramification filtration of the Galois
group of the field K in #(p) modulo its subgroup of commutators of order # p in
terms of generators of the group #(p).
Let K be a complete discrete valuation field of characteristic p > 0 with a finite
residue field k # F p N 0 . We assume that an uniformising element t 0 of the field K
is fixed and use the identification K = k((t 0 )). Choose a separable closure K sep
of the field K and set # = Gal(K sep /K). This group has the decreasing filtration
of normal subgroups {# (v)
} v>0 , which consists of higher ramification subgroups in
upper numbering, cf. [Se, Ch.III]. If #(p) is the Galois group of the maximal p­
extension of the field K, then #(p) is a free pro­p­group [Sh], and one can consider
the problem of description of the induced ramification filtration {#(p) (v)
} v>0 in
terms of generators of the group #(p). In this paper we apply methods from papers
[Ab1,2] to obtain this description modulo the closure C p (#(p)) of the subgroup of

  

Source: Abrashkin, Victor - Department of Mathematical Sciences, University of Durham

 

Collections: Mathematics