Discriminants of $\mathop{\mathfrak{S}}\nolimits_n$-orders

Stéphane R. Louboutin (1)

1 I2M - Institut de Mathématiques de Marseille

Let alpha be an algebraic integer of degree n >= 2. Let alpha(1),..., alpha(n) be the n complex conjugate of alpha. Assume that the Galois group Gal(Q(alpha(1),..., alpha(n))/Q) is isomorphic to the symmetric group S-n. We give a Z-basis and the discriminant of the order Z[alpha(1),..., alpha(n)]. We end up with an open question showing that this problem seems much harder when we assume that Q(alpha)/Q is already Galois or even cyclic.

Mots clés: algebraic integer; module; basis; discriminant

Domaines: Théorie des nombres [math.NT]

Stéphane R. Louboutin. Discriminants of $\mathop{\mathfrak{S}}\nolimits_n$-orders. International Journal of Number Theory, 2016, 12 (07), pp.1899 - 1905. ⟨10.1142/S1793042116501177⟩. ⟨hal-01288069⟩