Let $K$ be a field of characteristic different from 2 .

Show that if $L / K$ is an extension of degree 2 , then $L=K(x)$ for some $x \in L$ such that $x^{2}=a \in K$. Show also that if $L^{\prime}=K(y)$ with $0 \neq y^{2}=b \in K$ then $L$ and $L^{\prime}$ are isomorphic (as extensions of $K$ ) if and only $b / a$ is a square in $K$.

Now suppose that $F=K\left(x_{1}, \ldots, x_{n}\right)$ where $0 \neq x_{i}^{2}=a_{i} \in K$. Show that $F / K$ is a Galois extension, with Galois group isomorphic to $(\mathbb{Z} / 2 \mathbb{Z})^{m}$ for some $m \leqslant n$. By considering the subgroups of $\operatorname{Gal}(F / K)$, show that if $K \subset L \subset F$ and $[L: K]=2$, then $L=K(y)$ where $y=\prod_{i \in I} x_{i}$ for some subset $I \subset\{1, \ldots, n\}$.