Etudiant : BOUAOUINA HAMID

Filière : Master Mathématiques Pures (MMP)

Encadrant : Pr. EL FADIL LHOUSSAIN

Annèe : 2024

Résumé : The concept of prime ideal factorization is an important topic in algebraic number theory, traced back to the 19th century with the work of mathematicians, such as Kummer, as founder of the theory of ideals in algebraic number theory. We mostly consider number fields K = Q(θ) defined by a monic irreducible polynomial f(x) ∈ Z[x], where θ is a root of f(x), ZK its ring of integers, and p is a prime integer. The problem of the factorization of the ideal pZK is a major problem in algebraic number theory. In this manuscript, we try to gather results of mathematicians which answer various problems of factorization. In 1878 Dedekind gave a criterion to detect when p does not divide the index ind(θ) := (ZK : Z[θ]), and a procedure to construct the prime ideals of K dividing p in that case, in terms of the factorization modulo p of f(x). M. Bauer introduced an arithmetic version of Newton Polygons to construct prime ideals in cases when Dedekind’s criterion failed. This theory was developed and extended by Ø. Ore in his 1923 thesis and a series of papers that followed [16]. Ore’s work determines three successive factoriations of