Réciprocité et nombres premiers réguliers

`` Although it is not well known, Kummer at one time believed he had found a complete proof of Fermat's theorem...Seeking the best critic for his proof, Kummer sent his manuscript to Dirichlet...After a few days, Dirichlet replied with the opinion that the proof was excellent and certainly correct, provided the numbers in $\alpha$ could not only be decomposed into indecomposable factors, as Kummer proved, but that this could be done in only one way. If however, the second hypothesis couldn't be satisfied, most of the theorem for the arithmetic of numbers in $\alpha$ would be unproven and the proof of Kummer's theorem would fall apart. Unfortunately, it appeared to him that the numbers in $\alpha$ didn't actually possess this property in general. '' Kurt Hensel, Commemoration of the first centennial of Kummer's birth (1910) [Ribenboim 2].

Cet apologue est semblable à celui qui, en 1847, arriva à Lamé : lorsqu'il présenta sa démonstration du théorème de Fermat à l'Académie, il se vit aussitôt opposer une objection dirimante de Liouville qui fit remarquer une non unicité potentielle de la décomposition en facteurs irréductibles d'expressions polynomiales en racines énièmes de l'unité (confer Compte rendu des séances de l'Académie des Sciences, séance du 1^er mars 1847).