Corps biquadratiques monogènes

Abstract

Let\(K = \mathbb{Q}(\sqrt {dm} ,\sqrt {dn} )\) be a biquadratic number field (where d,m,n∈ℤ, are uniquely determined); we say that it is monogenic if its ring of integers OK is of the form ℤ[θ]. We show that K is monogenic if and only if the two following conditions are satisfied: (i) 2δm=2δn+4(2−δd) where δ=0 or 1 is defined by mn≡(−1)δ mod4; (ii) the equation (u2-v2)2(2δm)-(u2+v2)2(2δn)=±1 has solutions in ℤ.