confused about a statement of pure extension in Rotman's abstract algebra textbook

Question

I don't know how to prove the statement in the red rectangles. Namely, How to prove any field extension $K/k$ with $[K:k]\leq 2$ is a pure extension.

Answer 1

I assume we are not in characteristic $2$ and $[K:k]=2$. Let $v\in K\setminus k$ be any element. Then $K=k(v)$. As the extension is of degree $2$ and we are not in characteristic $2$ it is Galois. Let $\sigma$ be the non-trivial $k$-automorphism of $K$. Define $u=v-\sigma(v)$. Since we are not in characterstic $2$ we have $\sigma(u)\ne u$, hence $u\in K\setminus k$ and therefore $K=k(u)$. Furthermore we have $\sigma(u^2)=u^2$, so $u^2\in k$.