### Demushkin groups of uncountable rank

Algebra Seminar

12th March 2024, 4:00 pm – 5:00 pm

Fry Building, Room 2.04

Demushkin groups play an important role in number theory, being the maximal pro-p Galois groups of local fields. In 1996 Labute presented a generalization of the theory for countably infinite rank pro-p groups, and proved that the p-Sylow subgroups of the absolute Galois groups of local fields are Demushkin groups of infinite countable rank. These results were extended by Minac & Ware, who gave necessary and sufficient conditions for Demushkin groups of infinite countable rank to occur as absolute Galois groups.

In a joint work with Prof. Nikolay Nikolov, we extended this theory to Demushkin groups of uncountable rank. Since for uncountable cardinals there exist the maximal possible number of nondegenerate bilinear forms, the class of Demushkin groups of uncountable rank is much richer, and in particular, the groups are not determined completely by their invariants, in contrast to the countable case.

We present some results about the structure of Demushkin groups of uncountable rank, as well as equivalent conditions for being a Demushkin group, and investigate their ability to be realized as absolute Galois groups.

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