Splitting fields of pair of polynomials over finite fields

Question

Given $n \geq 1$, is it true that $x^{2^n} + 1 \in \mathbb{F}_p[x]$ splits over the splitting field of $x^{2^{n+1}} + 1 \in \mathbb{F}_p[x]$?

If so, how can I prove this? Hint preferred.

I tried using induction but ran into difficulties relating to irreducibility of the polynomials.

Thank you

Answer 1

Hint: If $a$ is a root of $x^{2^{n+1}}+1$, what can you say about $a^2$?

A full proof is hidden below.

If $a$ is a root of $x^{2^{n+1}}+1$, then $a^2$ is a root of $x^{2^{n}}+1$. Assuming $p\neq 2$, then, $x^{2^{n+1}}+1$ has $2^{n+1}$ distinct roots in its splitting field (since it is separable) and the squares of these roots give $2^n$ distinct roots of $x^{2^{n}}+1$. Thus $x^{2^{n}}+1$ splits over this splitting field. For $p=2$, the result is trivial since $x^{2^{n}}+1=(x+1)^{2^n}$ splits over $\mathbb{F}_2$.