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Soient K un corps de nombres et p un nombre premier ≥ 5. Notons μp le groupe des racines p-iemes de l'unite. On definit p comme etant K-regulier si p ne divise pas le nombre de classes du corps K(μp). Sous l'hypothese que p est K-regulier et inerte dans K, on etablit le second cas du theoreme de Fermat sur K pour l'exposant p. On utilise pour cela des arguments classiques, ainsi que le theoreme de Faltings selon lequel une courbe de genre au moins deux sur K n'a qu'un nombre fini de points K-rationnels. De plus, si K est un corps quadratique imaginaire, distinct de , on en deduit un enonce permettant souvent en pratique de demontrer le theoreme de Fermat sur K pour un exposant K-regulier donne. Mots-cles: Theoreme de Fermat; corps de nombres. Let K be a number field and p a prime number ≥5. Let us denote by μp 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(μp). 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 , we deduce a statement which allows often in practice to prove Fermat's Last Theorem over K for a given K-regular exponent.

Équation de Fermat et nombres premiers inertes