Discriminants of 𝔖n-orders

Abstract

Let [Formula: see text] be an algebraic integer of degree [Formula: see text]. Let [Formula: see text] be the [Formula: see text] complex conjugate of [Formula: see text]. Assume that the Galois group [Formula: see text] is isomorphic to the symmetric group [Formula: see text]. We give a [Formula: see text]-basis and the discriminant of the order [Formula: see text]. We end up with an open question showing that this problem seems much harder when we assume that [Formula: see text] is already Galois or even cyclic.