A note on Dedekind Criterion

Abstract

Let K = ℚ() be an algebraic number field with an algebraic integer having minimal polynomial f(x) over the field ℚ of rational numbers and AK be the ring of algebraic integers of K. For a fixed prime number p, let f (x) = ?1(x)e1⋯?r(x)er be the factorization of f(x) modulo p as a product of powers of distinct irreducible polynomials over ℤpℤ; with gi(x) ∈ℤ[x] monic. In 1878, Dedekind proved a significant result known as Dedekind Criterion which says that the prime number p does not divide the index [AK: ℤ;[]] if and only if Πi=1r? i(x)ei-1 is coprime with M̄(x) where M(x) = 1 p[f(x) - g1(x)e1⋯gr(x)er]. 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.