Let p>2 be a prime. Let \xi be a primitive p-root of unity. Let \Q(\xi) be the p-cyclotomic number field. Let r_p be the p-rank of the class group of \Q(\xi). Let x,y,z\in\Z-{0} be mutually co-prime. Let x^p+y^p+z^p=0 be the Fermat equation. Let t\in\Z, t\equiv -x/y mod p. Let \phi_{2m+1}(T), m=1,...,(p-3)/2 be the odd Mirimanoff Polynomials.

We give a canonical generalization of Mirimanoff congruences and an elementary proof that: if r_p< (p-1)/2, then \phi_{2m+1}(t)\equiv 0 mod p for at least (p-3)/2-r_p integers m, 1 \leq m \leq (p-3)/2.

We give several kinds of new summation criteria for first case of FLT . AS an example, the p-2 congruences : For each d\in\N, 1\leq d \leq p-2, if \sum_{l=1}^d \sum_{j=[(lp)/(d+1)]+1}^{[(lp)/d]} (1/j^3) \not\equiv 0 mod p, then the first case of FLT holds for p.

We prove the second case of FLT when p || y.

This paper rests on classical algebraic number theory without any connection with Wiles proof.

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Roland Queme <106104.1447@compuserve.com>