A note on Dedekind Criterion.

Let K = ℚ (𝜃) be an algebraic number field with 𝜃 an algebraic integer having minimal polynomial f (x) over the field ℚ of rational numbers and A K be the ring of algebraic integers of K. For a fixed prime number p , let f ̄ (x) = ḡ 1 (x) e 1 ⋯ ḡ r (x) e r be the factorization of f (x) modulo p as a product of powers of distinct irreducible polynomials over ℤ / p ℤ with g i (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 [ A K : ℤ [ 𝜃 ] ] if and only if ∏ i = 1 r ḡ i (x) e i − 1 is coprime with M ¯ (x) where M (x) = 1 p [ f (x) − g 1 (x) e 1 ⋯ g r (x) e r ]. 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. Algebra38 (2010) 684–696] using elementary valuation theory is given. [ABSTRACT FROM AUTHOR]