Critère effectif de puissancep-ième dans un corps de nombres galoisien
✍ Scribed by Georges Gras; Stéphane Jeannin
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 385 KB
- Volume
- 63
- Category
- Article
- ISSN
- 0022-314X
No coin nor oath required. For personal study only.
✦ Synopsis
Let e{0 be an integer of a Galois extension KÂQ (of degree n and Galois group G), and let p be a prime number. Let (=$ _ ) _ # G be a family of approximations (in C) of suitable p th roots = _ of the _(e), _ # G. Using a general criterion of the second author, we prove that there exists 2 (explicitly computable) such that if |=$ _ &= _ |<(1Ân) 2, for all _ # G, and | _ # G =$ _ &m| <2, for m # Z, m{0, then e is a p-power in K _ . Some consequences are given, such as the fact that the method of ``de vissages'' of cyclotomic units in real abelian fields (for the determination of fundamental units and class numbers) is now valid and considerably easier.