I'm trying to answer the following question:
"Suppose that $F=K(x_1,\dots,x_n)$ 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 $m\leq n$."
My first thought was show that $[F:K]=2^m$ and then find $2^m$ K-homs (and hence automorphisms) from $F$ to $F$ and thus the Galois group.
I'm wondering if this is the "right" approach, since it seems we need to define maps sending $x_i$ to $\pm x_i$, and then prove that these are homs by considering $F$ as a $K$-vector space, which, at first sight, appears messy and unenlightening.
Is there a conceptually clearer way of doing things here?
Thanks.