The first case of Fermat's Last Theorem for a prime exponent p can sometimes be proved using the existence of local obstructions. In 1823, Sophie Germain obtained an important result in this direction by establishing that, if 2p+1 is a prime number, the first case of Fermat's Last Theorem is true for p. In this paper, we investigate such obstructions over number fields. We obtain analogous results on Sophie Germain type criteria, for imaginary quadratic fields. Furthermore, extending a well known statement over ℚ, we give an easily testable condition which allows one occasionally to prove the first case of Fermat's Last Theorem over number fields for a prime number p ≡ 2 (mod 3).
2
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
Let p be an odd prime number. In this paper, we are concerned with the behaviour of Fermat curves defined over ℚ, given by equations $ax^{p} + by^{p} + cz^{p} = 0$, with respect to the local-global Hasse principle. It is conjectured that there exist infinitely many Fermat curves of exponent p which are counterexamples to the Hasse principle. This is a consequence of the abc-conjecture if p ≥ 5. Using a cyclotomic approach due to H. Cohen and Chebotarev's density theorem, we obtain a partial result towards this conjecture, by proving it for p ≤ 19.
3
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW