Irreducibility of $(n+1)+nx+(n-1)x^2+\ldots+x^n$

Asked Jun 20 '22 at 23:00

Active Jun 20 '22 at 23:01

Viewed 80 times

I'm trying to prove the irreducibility over the rationals of the polynomial defined by

$$f(x)=(n+1)+nx+(n-1)x^2\ldots+x^n$$

for all $n\in\mathbb{N}$. Computationally, I've verified it is always irreducible up to $n=4096$. It seems like it should be a polynomial that has been studied before, so I'm wondering if anyone recognizes it or if there's a simple trick I'm just missing. Any help would be appreciated!

edited Jun 20 '22 at 23:01

Gary

31,845

asked Jun 20 '22 at 23:00

Vibbz

Does this answer your question? Family of irreducible polynomials. There does not seem to be a definitive answer, but partial ones and some useful links. – dxiv Jun 20 '22 at 23:14

Irreducibility of $(n+1)+nx+(n-1)x^2+\ldots+x^n$

0 Answers0