Prime ideal decomposition in$F({}^m\sqrt \mu )$

Abstract

LetF be an algebraic number field and μ∈F such thatx ^m−μ is irreducible, wherem is an integer. Let$\mathfrak{p}$ be a prime ideal inF with$N (\mathfrak{p}) = pf$. The prime decomposition of$\mathfrak{p}$ in$F({}^m\sqrt \mu )$ is explicitly obtained in the following cases. Case 1:$\mu \mathop \equiv \limits^8 0 (\bmod \mathfrak{p}^a )$, (a,m) = 1 (where$\mu \mathop \equiv \limits^8 0 (\bmod \mathfrak{p}^a )$ means$\mu \equiv 0 (\bmod \mathfrak{p}^a )$,ς ≢ 0$(\bmod \mathfrak{p}^a + 1)$). Case 2:m ≡lt, wherel is a prime andl ≢ 0$(\bmod \mathfrak{p})$. Case 3:m ≢ 0$(\bmod \mathfrak{p})$ and every prime that dividesm also dividespf−1. It is not assumed that thev ^th roots of unity are inF for anyv ≠ 2.