The sum of two algebraic points is a new algebraic point

Question

Let $K / F$ be an extension of fields and let $x , y \in K$ be two algebraic points over $F$. I do not know if my proof of $x + y$ is an algebraic point over $F$ is correct. My attempt is the next: checking that $F(x , y) / F$ is a finite extension is not difficult: $$ [F(x , y) : F] = [(F(x))(y) : F(x)] [F(x) : F] $$ and as $[(F(x))(y) : F(x)]$ as $[F(x) : F]$ are finite because they are the degree of some irreducible polynomials. Since $F(x , y) / F$ is finite, then it is algebraic, which shows that $x + y$ is algebraic over $F$ because $x + y \in F(x , y)$.

Answer 1

Your proof is fine. It's the standard proof, once one has established that $\alpha \in K$ is algebraic over $F$ iff $[F(\alpha):F]$ is finite.

One detail, though. To be precise, write $$ [F(x , y) : F] = [F(x)(y) : F(x)] [F(x) : F] \le [F(y) : F] [F(x) : F] < \infty $$ noting that $[F(x)(y) : F(x)] \le [F(y) : F]$ because every equation for $y$ over $F$ is an equation for $y$ over $F(x)$.