Self-equivalences of the Gaussian field (Q1011123)

Self-equivalence of a number field \(K\) is a Hilbert-symbol equivalence of \(K\) with itself (for explanation what a Hilbert-symbol equivalence is, see the reviews of the earlier work of the author: [Can. J. Math. 57, No. 1, 180--203 (2005; Zbl 1073.11026)] or [Acta Arith. 121, No. 4, 327--334 (2006; Zbl 1115.11024)]. The wild set of a self-equivalence of \(K\) is the set of wild primes, that is, of finite primes \(P\) of \(K\) with the property that the parity of the \(P\)-order of elements of \(K\) is not preserved under the equivalence. Here the author proves that any finite set of primes of the Gaussian field \(\mathbb Q{(\sqrt{-1}\,)}\) occurs as the wild set of a self-equivalence of the field.