A Generalization of Dedekind Criterion

Abstract

Let K = ℚ(θ) be an algebraic number field with θ in the ring A K of algebraic integers of K and f(x) be the minimal polynomial of θ over the field ℚ of rational numbers. For a rational prime p, let (x) = (x) e 1 …(x) e r be the factorization of the polynomial (x) obtained by replacing each coefficient of f(x) modulo p into product of powers of distinct irreducible polynomials over ℤ/pℤ with g i (x) monic. In 1878, Dedekind proved that if p does not divide [A K :ℤ[θ]], then , where ℘1,…, ℘ r are distinct prime ideals of A K , ℘ i = pA K + g i (θ)A K with residual degree f(℘ i /p) = deg (x). He also gave a criterion which says that p does not divide [A K :ℤ[θ]] if and only if for each i, we have either e i = 1 or (x) does not divide where . The analog of the above result regarding the factorization in A K′ of any prime ideal 𝔭 of A K is in fact known for relative extensions K′/K of algebraic number fields with the condition “p ≠ | [A K :ℤ[θ]]” replaced by the assumption “every element of A K′ is congruent modulo 𝔭 to an element of A K [θ](†)”. In this article, our aim is to give a criterion like the one given by Dedekind which provides a necessary and sufficient condition for assumption (†) to be satisfied.