Let $F \leq E$ be an extension of fields, let $a \in E$ and let $b \in F$.
Show that $a$ is algebraic over $F$ if and only if $a + b$ is algebraic over $F$.
Is the best approach here to try and explicitly construct some polynomial $f(x) \in F[x]$ that satisfies $f(a+b)=0$ (to show the $\Rightarrow$ direction), or is it better to try and argue for the finiteness of $F(a+b)$? [I understand these are effectively the same thing].
And for the reverse direction ($\Leftarrow$), is it best to argue that, since $b \in F$, $F(b)=F$ and so $F(a+b)=F(a)$ (I'm not sure if this can simply be stated without justification) meaning that $|F(a):F|=|F(a+b):F|=n$, where n is the degree of the minimmum polynomial for $a+b$, which exists by assumption.
Thinking about another problem
How can we change approach to deal with the equivalent problem but with $ab$ rather than $a+b$. Does the $\iff$ relation still hold? (We should probably add the assumption that $b \neq 0 $ for this case).
Any help is very appreciated. Thanks!