Is $x^{2^{n+1}} - x^{2^n} + 1$ is irreducible over the integers for all $n$?

Question

It seems $x^2-x+1$ or $x^4-x^2+1$ are irreducible over the integers.

Is $x^{2^{n+1}} - x^{2^n} + 1$ is irreducible for all non-negative integer $n$? If so, how to prove?

Answer 1

Multiplying by $ (X^{2^n} + 1) $, we have that

$$ (X^{2^n} + 1)(X^{2^{n+1}} - X^{2^n} + 1) = X^{3 \cdot 2^n} + 1 $$

It is clear from this that the roots of $ X^{2^{n+1}} - X^{2^n} + 1 $ are the primitive $ 3 \cdot 2^{n+1} $th roots of unity, and the degree of the corresponding minimal polynomial is $ \varphi(3 \cdot 2^{n+1}) = 2^{n+1} $. Thus, $ X^{2^{n+1}} - X^{2^n} + 1 $ is the $ 3 \cdot 2^{n+1} $th cyclotomic polynomial, and it follows that it is irreducible.