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- Title
A note on Dedekind Criterion.
- Authors
Jakhar, Anuj; Khanduja, Sudesh K.
- Abstract
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.
- Subjects
ALGEBRAIC numbers; DEDEKIND sums; PRIME numbers; RATIONAL numbers; RING theory; ALGEBRAIC fields; IRREDUCIBLE polynomials; POLYNOMIAL rings
- Publication
Journal of Algebra & Its Applications, 2021, Vol 20, Issue 04, pN.PAG
- ISSN
0219-4988
- Publication type
Article
- DOI
10.1142/S0219498821500663