= Closure of separable algebraic elements under addition
{title2=$\alpha,\beta\text{ separable over }F\Longrightarrow\alpha+\beta\text{ separable over }F$}
If $\alpha$ and $\beta$ are <separable algebraic elements>, put $K=F(\alpha)$ and $L=K(\beta)$. The <minimal polynomial> of $\beta$ over $K$ divides its <separable polynomial> over $F$, and is therefore separable. The <extension count for field embeddings> gives $[K:F]$ embeddings of $K$ into an <algebraic closure>, each with $[L:K]$ extensions to $L$. Hence $L$ has $[L:F]$ embeddings. For $E=F(\alpha+\beta)$, each embedding of $E$ has at most $[L:E]$ extensions, so there are at least $[E:F]$ embeddings of $E$ and hence exactly that number. Since $E$ is a simple <field extension>, its <minimal polynomial> has $[E:F]$ distinct roots. Thus $\alpha+\beta$ is a <separable algebraic element>.
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