On a theorem of Dedekind

Abstract

Let K = Q(θ ) be an algebraic number field with θ in the ring AK 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 ƒ^-(x) = g^-1(x)^e1 ···g^-r(x)^er be the factorization of the polynomial ƒ^-(x) obtained by replacing each coefficient of f(x) modulo p into product of powers of distinct monic irreducible polynomials over Z/pZ: Dedekind proved that if p does not divide [A_K : Z[θ ]]; then the factorization of pAK as a product of powers of distinct prime ideals is given by pA_K = p^e1 ₁ ···p^er _r ; with pi = pA_K + g_i(θ )A_K; and residual degree ƒ(p_i=p) = deg g^-i(x): In this paper we prove that if the factorization of a rational prime p in AK satisfies the above mentioned three properties, then p does not divide [A_K : Z[θ ]]: Indeed the analogue of the converse is proved for general Dedekind domains. The method of proof leads to a generalization of one more result of Dedekind which characterizes all rational primes p dividing the index of K.