Abstract
AbstractIn this article we examine conditions for the appearance or nonappearance of the twoextra-special 2-groups of order 32 as Galois groups over a fieldFof characteristic not 2. The groups in question are the central productsDDof two dihedral groups of order 8, andDQof 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 overFimplies 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 ofDDandDQ. We conclude by providing an explicit construction of field extensions with Galois groupDD.
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