Irreducibility of polynomials $x^{2^{n}}+1$

Question

I would like to if the polynomials of the form $x^{2^{n}}+1$ are irreducible over $\mathbb{Q}$ and in that case if there is some "easy" proof for that (where easy means not using a big theory like Galois).

Answer 1

Let $f(x)=x^{2^n}+1$. Note that if $f(x+1)$ is irreducible, then so is $f(x)$.

We have:

$$f(x+1)=\left(x^{2^n} + \binom{2^n}{1}x^{2^n-1}+\dots+\binom{2^n}{2^n-1}x+1\right)+1$$

Note that $2$ divides all the coefficients except that of $x^{2^n}$, and $4$ does not divide the constant coefficient, $2$. Thus the polynomial is irreducible by Eisenstein's criterion.