Galois group of $x^{2^k}+1$

Question

What is the Galois group of $f(x)=x^{2^k}+1$ over $\mathbb{Q}$?

Answer 1

The splitting field of $f(x)$ over $\mathbb Q$ is $\mathbb Q (\zeta_{2^{k+1}})$, where $\zeta_{2^{k+1}}$ is the $2^{k+1}$-th primitive root of unit. When $k=0$ we have that $\zeta_{2}=-1$, so $\mathbb Q (-1)=\mathbb Q$ and $Gal(\mathbb Q/\mathbb Q)=\{e\}$.When $k\geq 1$ it's known that $Gal(\mathbb Q(\zeta_{2^{k+1}})/\mathbb Q) \cong (\mathbb Z_{2^{k+1}})^*\cong \mathbb Z_{2^{k-1}} \times \mathbb Z_2$.