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On vectorial polynomials and coverings in characteristic 3. (English) Zbl 1092.12006

Let \(K\) be a field containing the field \(F_9\) of nine elements, let \(2S_4\) denote a double cover of the symmetric groups \(S_4\), let \(H\) denote either the quaternion group \(Q_8\) of order \(8\) or the dihedral group \(D_8\) of order \(8\), and let \(*\) denote a central product. This paper provides an explicit description of the entire family of Galois extensions of \(K\) with Galois groups \(2S_4*H\) and determines the disciminants of these extensions. The method of construction combines Abhyankar’s embedding criterion and Serre’s trace formula. The determination of the discriminant is a step towards local uniformization for three-dimensional varieties in positive characteristic.

MSC:

12F12 Inverse Galois theory
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