Équation de Fermat et nombres premiers inertes

2014/11/27 by Kraus, Alain
#11D41 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1411.7537

Abstract

Let K be a number field and p a prime number ≥ 5. Let us denote by μ_p the group of the pth roots of unity. We define p to be K-regular if p does not divide the class number of the field K(μ_p). Under the assumption that p is K-regular and inert in K, we establish the second case of Fermat's Last Theorem over K for the exponent p. We use in the proof classical arguments, as well as Faltings' theorem stating that a curve of genus at least two over K has a finite number of K-rational points. Moreover, if K is an imaginary quadratic field, other than \bf Q(√(-3)), we deduce a statement which allows often in practice to prove Fermat's Last Theorem over K for the K-regular exponents.

Related

• Fermat's Last Theorem for regular primes
• Sur le théorème de Fermat sur \bf Q(√(5))
• Remarques sur le premier cas du théorème de Fermat sur les corps de nombres
• Fermat's Last Theorem over \mathbb Q(√(5)) and \mathbb Q(√(17))
• A Study of Kummer's Proof of Fermat's Last Theorem for Regular Primes