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On the factors of a polynomial
Author(s) -
Jakhar Anuj
Publication year - 2020
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms.12315
Subject(s) - mathematics , irreducibility , irreducible polynomial , prime (order theory) , combinatorics , degree (music) , polynomial , prime power , field (mathematics) , minimal polynomial (linear algebra) , cyclotomic polynomial , discrete mathematics , reciprocal polynomial , pure mathematics , matrix polynomial , mathematical analysis , physics , acoustics
In this article, we show that if f ( x ) = a n x n + a n − 1x n − 1 + ⋯ + a 0 , a 0 ≠ 0 is a polynomial with rational coefficients and if there exists a prime p whose highest power r i dividing a i (wherer i = ∞ ifa i = 0 ) satisfiesr n = 0 , n r i ⩾ ( n − i ) r 0 > 0 for 0 ⩽ i ⩽ n − 1 , then f ( x ) has at most gcd ( r 0 , n ) irreducible factors over the field Q of rational numbers and each irreducible factor has degree at least n / gcd ( r 0 , n )over Q . This result extends the famous Eisenstein–Dumas irreducibility criterion. In fact, we prove our result in a more general setup for polynomials with coefficients in arbitrary valued fields.