Irreducibility and Galois groups of truncated binomial polynomials
Article Type
Research Article
Publication Title
International Journal of Number Theory
Abstract
For positive integers n ≥ m, let Pn,m(x):=σj=0mn j xj = n 0 + n 1 x + ... + n mxm be the truncated binomial expansion of (1 + x)n consisting of all terms of degree ≤ m. It is conjectured that for n > m + 1, the polynomial Pn,m(x) is irreducible. We confirm this conjecture when 2m ≤ n < (m + 1)10. Also we show for any r ≥ 10 and 2m ≤ n < (m + 1)r+1, the polynomial Pn,m(x) is irreducible when m ≥max{106, 2r3}. Under the explicit abc-conjecture, for a fixed m, we give an explicit n0,n1 depending only on m such that ∀ n ≥ n0, the polynomial Pn,m(x) is irreducible. Further ∀ n ≥ n1, the Galois group associated to Pn,m(x) is the symmetric group Sm.
First Page
1663
Last Page
1680
DOI
10.1142/S1793042124500817
Publication Date
7-1-2024
Recommended Citation
Laishram, Shanta and Yadav, Prabhakar, "Irreducibility and Galois groups of truncated binomial polynomials" (2024). Journal Articles. 4864.
https://digitalcommons.isical.ac.in/journal-articles/4864