Extra-Special Groups of Order 32 as Galois Groups

Smith, Tara L.

Canadian journal of mathematics, Tome 46 (1994) no. 4, pp. 886-896

Résumé

In this article we examine conditions for the appearance or nonappearance of the two extra-special 2-groups of order 32 as Galois groups over a field F of characteristic not 2. The groups in question are the central products DD of two dihedral groups of order 8, and DQ of a dihedral group with the quaternion group, obtained by identifying the central elements of order 2 in each factor group. It is shown that the realizability of each of these groups as Galois groups over F implies the realizability of other 2-groups (which are not their quotient groups), and in turn that realizability of certain other 2-groups implies the realizability of DD and DQ. We conclude by providing an explicit construction of field extensions with Galois group DD.

Détail

DOI : 10.4153/CJM-1994-050-2

Mots-clés : 12F

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