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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-\(p\) Demu┼íkin 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)

Citations:

Zbl 0857.12002
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