Choose a basis $b_1, \ldots, b_k$ for $L$ such that $b_1 \in k$
Then consider the map $\phi: b_i \to -b_i$ for $i > 1$
This definitely fixes $k$ pointwise, now I want to show that this is a field automorphism.
Let's pretend I did that already. Then given $z\in L$, we should have $(x-z)(x-\phi(z)) = x^2 - x(z + \phi(z)) - z\phi(z)$
I guess we need that $z\phi(z) \in k$ even then
Oh wait I see how we can use finite maybe