Prove $x^n+1$ is irreducible over $\mathbb{Q}[X]$ iff $n=2^k$ for $k \in \mathbb{N}$

Question

Unfortunately, I cannot find any information on or anything similar to this particular question. Might be quite new.

In all honesty, I don't know how to tackle either side of this question. By induction? By contradiction?

The theorems I just learned are Eisenstein"s criterion and reduction by modulo $n$ but nothing else. And both do not seem to help in any explicit way.

Does anyone have ideas on how to solve this problem at all? Thank you

Answer 1

1. If $n$ is divisible by an odd positive integer $d$, then $x^n+1$ is divisible by $x^{n/d}+1$.
2. If $f(x) = x^{2^k}+1$, apply Eisenstein to $f(x+1)$.