## Small maximal pro-$$p$$ Galois groups.(English)Zbl 0902.12003

Let $$p$$ be an odd prime, and let $$K$$ be a field of characteristic not $$p$$ and containing a primitive $$p$$th root of unity. Let $$G_K(p)$$ be the maximal pro-$$p$$ Galois group of $$K$$, i.e. the Galois group over $$K$$ of the field $$K(p)$$ which is the composite inside a fixed algebraic closure of all finite Galois $$p$$-extensions of $$K$$. This paper provides a classification of all such groups which are (topologically) generated by at most $$4$$ elements, with the exception of the pro-$$p$$ Demuškin groups on $$4$$ generators. A similar classification is already known for $$p = 2$$ and up to $$5$$ generators, via the classification of abstract Witt rings with at most $$2^5$$ square classes. The computations in this paper are based on classifying $$p$$-quaternionic pairings of small order; these pairings were introduced by Y. Hwang and B. Jacob [Can. J. Math. 47, 527-543 (1995; Zbl 0857.12002)].

### MSC:

 12F10 Separable extensions, Galois theory 12G05 Galois cohomology 13K05 Witt vectors and related rings (MSC2000)

Zbl 0857.12002
Full Text: