I know that if $K$ is an extension of $\mathbb{Q}$ of degree $2$, then $K = \mathbb{Q}(\sqrt{d})$ for some squarefree integer $d$. I understand that this is not the case for degree $2$ extensions of $\mathbb{R}$.
What can we say for degree $2$ extensions of fields of characteristic $p\in$ prime?
Given a field $k$ not of characteristic $2$, and given $[K:k]=2$, let $K=k(\beta)$ with $\beta$ a zero of $x^2+ax+b$. Replacing $x$ by $x-{a\over 2}$ converts this to the form $x^2-b'$. Thus, the extension is obtained by adjoining $\sqrt{b'}$.
In characteristic $2$, extensions obtained by adjoining square roots are inseparable, but they are still there.
In characteristic $2$, the separable quadratic extensions are obtained by adjoining zeros of Artin-Schreier polynomials $x^2-x+a$.
Similarly, in characteristic $p$, separable degree-$p$ extensions are obtained by adjoining zeros of Artin-Schreier $x^p-x+a$.
