Galois group of $x^2-2$ is $\mathbb{Z}/2\mathbb{Z}$

Compute the Galois group of $x^2-2$.

Solution

Over $\QQ$, the roots of $x^2-2$ are $\pm\sqrt2$, and $\sqrt2\notin\QQ$, so $x^2-2$ is irreducible and its splitting field is $\QQ(\sqrt2)$, of degree $2$ over $\QQ$. A group of order $2$ is cyclic, so \[\Gal(\QQ(\sqrt2)/\QQ)=\{\mathrm{id},\ \sqrt2\mapsto-\sqrt2\}\cong\boxed{\ZZ/2\ZZ}.\]