Construction of 2-ADIC Galois extension with wild inertia given by an extra special 2-group

Given any field F and an odd integer n, suppose K be a degree 2n1 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 T0 { namely, the multiplicative group generated by basis elements of the even Clifford algebra associated with the quadratic form X2 1 + ::: + X2 n. 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) = T0.