Tècniques geomètriques en monogeneïcitat

Abstract

By the primitive element theorem, any number field K of degree n can be written as Q(α) for some α in K. However, the analogous affirmation is not always true in the case of the ring of integers. When the ring of integers of K is Z[α], we say K is monogenic. Every cubic number field determines a non-trivial F3-orbit in H^1(Q,E[φ]), where E is the elliptic curve and φ is a certain 3-isogeny. In this work, we review the proof of this fact and use it to obtain bounds on the number of monogenic cubic number fields of discriminant D in terms of the Mordell-Weil group of E^D : Y^2 = 4X^3+D. We also compute a general expression for the cocycle associated to any pure cubic number field of Dedekind type I, which we use to characterize the sum of two such cocycles.