$x^{2n} + x^{2n-1} + x^ {2n-2} +\ldots+ x + 1$ is irreducible for any $n\in \mathbb N$ in $F_2[x]$. True or false?

Question

Will the polynomials of the following set $A$ be irreducible in $F_2[x]$?

$A = [x^{2n} + x^{2n-1} + x^ {2n-2} + \ldots+ x + 1 : n\in \mathbb N]$

Can anyone please give me hints how to proceed?

Every term is there. I meant every polynomial of degree $2n$ has $2n+1$ terms.

Answer 1

It's false in general, for example for $n=4$ : $x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1 = (x^2+x+1)(x^6+x^3+1)$.

In general $\frac{x^n-1}{x-1}$ is irreducible iff n is prime(the cyclotomic poly) over $ \mathbb{Q}[X]$