Problem 7

Let $F$ be a field and $E$ be an extension field of $F$.

1. Let $a_{0},a_{1},\dots,a_{n}\in E$ be algebraic over $F$. Prove that the zero(es) of the polynomial

   $$f(x)=a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n}x^{n}$$

   are also algebraic over $F$.

2. Deduce that if $\alpha$ and $\beta$ are both transcendental over $F$, then either $\alpha+\beta$ or $\alpha\beta$ is also transcendental.