Summary: Towards arithmetical description of the Galois group of a local field.
Victor Abrashkin (Durham University)
Let K be a complete discreet valuation field with residue field of characteristic
p. Let #K be the absolute Galois group of K and let #K (p) be the Galois group of
the maximal p­extension of K. What we can say about its structure? The known
cases are:
--- if char K = p then #K (p) is pro­p­free;
--- if char K = 0 and k is finite (i.e. K is one­dimensional), then
--- if # p /
# K, then #K (p) is pro­p­free;
--- if # p # K, then #K (p) is the Demushkin group.
What we can say if K is an N­dimensional local field? (This means that k is
an (N - 1)­dimensional local field.) Notice that ``1­dimensional'' methods can't be
directly generalized to study the higher dimensional case.
Alternative ``1­dimensional'' approach: characteristic p case.
Assume for simplicity that K = F p ((t)). Then the Artin­Schreier theory gives
an explicit description of the group #K /# p
K C 2 (#K ), where C 2 (#K ) is the closed
subgroup generated by commutators of order # 2. This group appears as the Galois
group of the extension K({T a | (a, p) = 1 or a = 0})/K, where T p

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

Collections: Mathematics