A note on Dedekind Criterion

Abstract

Let [Formula: see text] be an algebraic number field with [Formula: see text] an algebraic integer having minimal polynomial [Formula: see text] over the field [Formula: see text] of rational numbers and [Formula: see text] be the ring of algebraic integers of [Formula: see text]. For a fixed prime number [Formula: see text], let [Formula: see text] be the factorization of [Formula: see text] modulo [Formula: see text] as a product of powers of distinct irreducible polynomials over [Formula: see text] with [Formula: see text] monic. In 1878, Dedekind proved a significant result known as Dedekind Criterion which says that the prime number [Formula: see text] does not divide the index [Formula: see text] if and only if [Formula: see text] is coprime with [Formula: see text] where [Formula: see text]. This criterion has been widely used and generalized. In this paper, a simple proof of Generalized Dedekind Criterion [S. K. Khanduja and M. Kumar, On Dedekind criterion and simple extensions of valuation rings, Comm. Algebra 38 (2010) 684–696] using elementary valuation theory is given.