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- W1622534802 abstract "Let p be an odd prime. Let K = Q(zeta) be the p-cyclotomic number field. Let v be a primitive root mod p and sigma : zeta --> zeta^v be a Q-isomorphism of the extension K/Q generating the Galois group G of K/Q. For n in Z, the notation v^n is understood by v^n mod p with 1 leq v^n leq p-1. Let P(X) = sum_{i=0}^{p-2} v^{-i}X^i in Z[X] be the Stickelberger polynomial. P(sigma) annihilates the class group C of K. There exists a polynomial Q(X) in Z[X] such that P(sigma)(sigma-v) = ptimes Q(sigma) and such that Q(sigma) annihilates the p-class group C_p of K (the subgroup of exponent p of C). In the other hand sigma^{(p-1)/2}+1 annihilates the relative class group of K. The simultaneous application of these results brings some informations on the structure of the class group C, give some explicit congruences in Z[v] mod p for the p-class group C_p of K and some explicit congruences in Z[v] mod h for the h-class group of K for all the prime divisors h not = p of the class number h(K). We detail at the end the case of class number of quadratic and biquadratic fields contained in the cyclotomic field K and give a general MAPLE algorithm." @default.
- W1622534802 created "2016-06-24" @default.
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- W1622534802 date "2006-09-26" @default.
- W1622534802 modified "2023-09-27" @default.
- W1622534802 title "On prime factors of class number of cyclotomic fields" @default.
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