Etage initial d'une ℤ-extension

Abstract

For a number fieldF that contains ζl a lth root of unity (l is a prime number), we determine thex such thatF\((\sqrt[\ell ]{x})\) can be embedded in a ℤl-extension. We approach the corresponding Kummer radical with the notion of being locally everywhere embedded in a ℤl-extension. An idelic description of Galois group is appropriate especially as we utilize the l-adic group of idele of [15]. The illustration concerns l=3 and biquadratic field ℚ\((\zeta _3 ,\sqrt d )\). We detail the step of the calculus and fournish numerical tables.