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00925njm 22002777a 4500 dc Pedret Martínez, Francesc author 2023-10-17 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. Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Teoria de nombres Algebraic number theory Elliptic curves Algebraic number theory number fields monogenicity power integral bases index form elliptic curves Selmer groups Tate-Shafarevich group Weil-Châtelet group cubic rings binary cubic forms Nombres, Teoria algebraica de Corbes el·líptiques Classificació AMS::11 Number theory::11R Algebraic number theory: global fields Classificació AMS::14 Algebraic geometry Tècniques geomètriques en monogeneïcitat