Idéaux premiers totalement décomposés et sommes de Newton

2020/12/10 by Bernardi, Dominique, Kraus, Alain
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2012.05569

Abstract

Let K be a number field and f∈ K[X] an irreducible monic polynomial with coefficients in O_K, the ring of integers of K. We aim to enounce an effective criterion, in terms of the Galois group of f over K and a linear recurrence sequence associated to f, allowing sometimes to characterize the prime ideals of O_K modulo which f completely splits. If α is a root of f, this criterion therefore gives a characterization of the prime ideals of O_K which split completely in K(α). It does apply if the degree of f is at least 4 and the Galois group of f is the symmetric group or the alternating group.

Related

• On Splitting Types, Discriminant Bounds, and Conclusive Tests for the Galois Group
• A note on prime divisors of polynomials P(T^k), k ≥ 1
• A new computational approach to ideal theory in number fields
• Irreducibility criterion, irreducible factors, Newton polygon techniques
• The ring of finite algebraic numbers and its application to the law of decomposition of primes