Explicit Form of Cassels’ p-adic Embedding Theorem for Number Fields

Abstract

AbstractIn this paper, we give a general explicit form of Cassels’ p-adic embedding theorem for number fields. We also give its reûned form in the case of cyclotomic fields. As a byproduct, given an irreducible polynomial f over ℤ, we give a general unconditional upper bound for the smallest prime number p such that f has a simple root modulo p.