Determination of the orders generated by a cyclic cubic unit that are Galois invariant

Jun Ho Lee (1) , Stéphane R. Louboutin (2)

1 KIAS - Korea Institute for Advanced Study
2 I2M - Institut de Mathématiques de Marseille

Let ϵ be a totally real cubic algebraic unit. Assume that the cubic number field Q(ϵ)Q(ϵ) is Galois. In this situation, it is natural to ask when the cubic order Z[ϵ]Z[ϵ] is invariant under the action of the Galois group Gal(Q(ϵ)/Q)Gal(Q(ϵ)/Q). It seems that this natural problem has never been looked at. We give an answer to this problem (e.g., we show that if ϵ is totally positive, then this happens in only 12 cases).

Mots clés: cubic unit; cubic orders; fundamental units

Domaines: Théorie des nombres [math.NT]

Jun Ho Lee, Stéphane R. Louboutin. Determination of the orders generated by a cyclic cubic unit that are Galois invariant. Journal of Number Theory, 2015, 148, pp.33-39. ⟨10.1016/j.jnt.2014.09.031⟩. ⟨hal-01287819⟩