Given any field F and an odd integer n, suppose K be a degree 2n --1 multiquadratic extension of F. We consider the conditions under which there is a Galois extension E of F such that Gal(E/F) is a particular extra special 2-group Gamma0 -- namely, the multiplicative group generated by basis elements of the even Clifford algebra associated with the quadratic form X21+&ldots;+X2n . These conditions can be restated in terms of the Weil index, which can be computed explicitly as a Gauss sum when F = Q2n . We prove an equidistribution of Gauss sums for quadratic characters on Q2n of conductor 4Z2n . As a consequence, we prove that, when n is an odd prime greater than 3, there exists a Galois extension K of Q2 such that K is a multiquadratic extension of Q2n that admits a quadratic extension E such that Gal( E/F) ≅ Gamma0.
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