2026-04-17T08:11:45Zhttps://recercat.cat/oai/request

oai:recercat.cat:2117/3959972025-07-23T02:22:41Zcom_2072_1033col_2072_452951

urn:hdl:2117/395997 Tècniques geomètriques en monogeneïcitat Pedret Martínez, Francesc 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 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. 2023-10-17 Master thesis http://creativecommons.org/licenses/by/4.0/ Open Access Attribution 4.0 International Universitat Politècnica de Catalunya