Small maximal pro-\(p\) Galois groups
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 \textit{Y. Hwang} and \textit{B. Jacob} [Can. J. Math. 47, 527-543 (1995; Zbl 0857.12002)].
scientific article; zbMATH DE number 1151666