On the irreducible factors of a polynomial II

Jakhar, Anuj; Kotyada, Srinivas

doi:10.1016/j.jalgebra.2020.02.045

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2021

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“…For example, consider the polynomial φ(x) = x 2 − x + 11 which is irreducible modulo 2, 3 and 5. Then the polynomial f (x) = φ(x) 4 15 + 6 φ(x) 2…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

See 1 more Smart Citation

A Useful Irreducibility Result for Integer Polynomials

Jakhar

2019

The American Mathematical Monthly

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Let c be a fixed integer such that c ∈ {0, 2}. Let n be a positive integer such that either n ≥ 2 or 2n + 1 = 3 u for any integer u ≥ 2 according as c = 0 or not. Let φ(x) belonging to Z[x] be a monic polynomial which is irreducible modulo all primes less than 2n + c. Let a i (x) with 0 ≤ i ≤ n − 1 belonging to Z[x] be polynomials having degree less than deg φ(x). Let a n ∈ Z and the content of (a n a 0 (x)) is not divisible by any prime less than 2n + c. For a positive integer j, if u j denotes the product of the odd numbers ≤ j, then we show that the polynomial an u2n+c φ(xu2j+c is irreducible over the field Q of rational numbers. This generalises a well-known result of Schur which states that the polynomial n j=0 a j x 2j u2j+c with a j ∈ Z and |a 0 | = |a n | = 1 is irreducible over Q. We illustrate our result through examples.

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“…For example, consider the polynomial φ(x) = x 2 − x + 11 which is irreducible modulo 2, 3 and 5. Then the polynomial f (x) = φ(x) 4 15 + 6 φ(x) 2…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

“…It may be pointed out that, in the above examples, the irreducibility of the polynomials f 5 (x) and g 4 (x) do not seem to follow from any known irreducibility criterion (cf. [1], [2], [3], [4], [5], [6] and [7]).…”

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confidence: 99%