Does $\alpha$ need to be transcendental over F?

Question

In the book there is this exercise:

Let E be an extension fiel of F, with $\alpha, \beta \in E$. Suppose $\alpha$ is transcendental over F but algebraic over $F(\beta)$. Show that $\beta$ is algebraic over $F(\alpha)$.

The suggested solution was:

It doesn't seem that they use that $\alpha$ is transcendental in F, or do they? Or is the statement valid without this assumption?

Answer 1

Actually, they use that hypothesis in the part

a polynomial in $\alpha$ with coefficients that are polynomials in $\beta$ can be formally rewritten as a polynomial in $\beta$ with coefficients that are polynomials in $\alpha$

to conclude that this new polynomial (seen in $(F[\alpha])[x]$) is not the zero polynomial. This is necessary to conclude that $\beta$ satisfy a non-zero polynomial over $F(\alpha)$ so $\beta$ is algebraic over $F(\alpha)$.