Effective criterion of power p in a Galois number field

[摘要] Let e not equal 0 be an integer of a Galois extension K/Q (of degree n and Galois group G), and let p be a prime number. Let (epsilon'(sigma))(sigma is an element of G) be a family of approximations (in C) of suitable pth roots epsilon(sigma) of the sigma(e), sigma is an element of G. Using a general criterion of the second author, we prove that there exists Delta (explicitly computable) such that if /epsilon'(sigma) - epsilon sigma/ < (1/n) Delta, for all sigma is an element of G, and /Sigma(sigma is an element of G) epsilon'(sigma) - m/ < Delta, for m is an element of Z, m not equal 0: then e is a p-power in K-x. Some consequences are given, such as the fact that the method of ''devissages'' of cyclotomic units in real abelian fields (for the determination of fundamental units and class numbers) is now valid and considerably easier. (C) 1997 Academic Press.

[发布日期] 1997-04-01