Show that a Galois extension is always generated by a square in characteristic 0

Question

Suppose that $char(K) = 0$ and that $L/K$ is a Galois extension. I have showed that $L = K(\alpha)$ for some $\alpha \in L^*$ and that $L\neq K(\alpha^2)$ iff $\exists \sigma \in Gal(L/K)$ such that $\sigma(\alpha) = -\alpha$ Now I only need to show that there exists $\beta \in L$ such that $L = K(\beta^2)$. I am guessing that $\beta = (\alpha + 1)$ but I'm not sure if this is true or how would you prove it.