The construction of the Hilbert genus fields of real cyclic quartic fields

Abstract

Let k be a number field and let H(k) denote the Hilbert class field of k, that is the maximal abelian unramified extension of k. It is known by class field theory that the Galois group of the extension H(k)/k, i.e., G := Gal(H(k)/k), is isomorphic to Cl(k), the class group of k (cf. [13, p. 228]). The Hilbert genus field of k, denoted by E(k), is the invariant field of G. Thus, by Galois theory, we have: Cl(k)/Cl(k) ≃ G/G ≃ Gal(E(k)/k),