I've read the following statement in Manin-Tsfasman's survey, Proposition 1.1.1 (I'm sorry I could only find the link in Russian):
Let $X$ be a smooth, projective rational curve over a field $k$. Then $-K_X$ gives an embedding $X\hookrightarrow \Bbb{P}^2$ of $X$ as a degree 2 curve on $\Bbb{P}^2$.
This seems like a classical result, but I don't know how this can be shown.
When $k$ is algebraically closed, I know how to prove even more. Take a point $P\in X$. Since $\deg(P)=1\geq 2g(X)=0$, then by Hartshorne's Corollary 3.2, chapter IV, we have an embedding $X\hookrightarrow \Bbb{P}^1$, which proves $X$ is actually isomorphic to $\Bbb{P}^1$.
How do we approach this when $k$ is arbitrary?