Here is my proof for the statement that the sum of two algebraic numbers is algebraic:
Let $\alpha,\beta$ be algebraic numbers.
Then we know that $[\Bbb Q(\alpha):\Bbb Q]$ is finite. Similarly, $[\Bbb Q(\beta):\Bbb Q]$ is also finite. Let $f \in \Bbb Q[x]$ be the minimal polynomial of $\beta$ over $\Bbb Q.$ Then $f \in (\Bbb Q(\alpha))[x]$ statisfying $f(\beta)=0.$This means $[\Bbb Q(\alpha,\beta):\Bbb Q(\alpha)]$ is finite. Therefore $[\Bbb Q(\alpha,\beta):\Bbb Q]$ is finite. Therefore $\Bbb Q(\alpha,\beta)/\Bbb Q$ is a finite extension. Therefore $\Bbb Q(\alpha+\beta)/\Bbb Q$ is also a finite extension(hence an algebraic extension). Therefore $\alpha+\beta$ is algebraic.
Can someone check whether my proof for the statement is correct or not? Thanks.