When does conjugation in simple field extensions coincide with the identity?

Question

Suppose I have a field $F$ and a simple extension of $F$, $F(\alpha)$, and let's assume that $\alpha^2 \in F$. Now take the "conjugation" automorphism $\sigma:F(\alpha) \to F(\alpha)$, $\sigma(a + b\alpha) = a-b\alpha$ for $a, b \in F$. My question is, does this map $\sigma$ ever coincide with the identity map? My guess is that for this to happen, we would need $\sigma(\alpha) = -\alpha = \alpha$ somehow. So every element is its own additive inverse, so maybe something about characteristic of $F$ being $2$. Is there a more structured way to think about this?

Answer 1

That's already it. You need that $2=0$ to make $-\alpha=\alpha$. This is called a purely inseparable extension. Perhaps you also count the case that already $\alpha\in F$ and the extension is not really an extension to begin with.