Let $L / K$ be a field extension.

(a) State what it means for $\alpha \in L$ to be algebraic over $K$, and define its degree $\operatorname{deg}_{K}(\alpha)$. Show that if $\operatorname{deg}_{K}(\alpha)$ is odd, then $K(\alpha)=K\left(\alpha^{2}\right)$.

[You may assume any standard results.]

Show directly from the definitions that if $\alpha, \beta \in L$ are algebraic over $K$, then so too is $\alpha+\beta$.

(b) State what it means for $\alpha \in L$ to be separable over $K$, and for the extension $L / K$ to be separable.

Give an example of an inseparable extension $L / K$.

Show that an extension $L / K$ is separable if $L$ is a finite field.