I am starting to study field theory and I encountered this question :
Suppose that $L:K$ is an extension, that $\alpha$ is an element of L which is
transcendental over K, and that $f$ is a non-constant element of
$K[x]$. Show that $f(\alpha)$ is transcendental over $K$. Show that, if $\beta$ is an
element of L which satisfies $f(\beta) =\alpha$, then $\beta$ is transcendental over
$K$.
I have already shown that $f(\alpha)$ is transcendental over $K$ but I'm having trouble showing that $\beta$ is transcendental. Any help would be appreciated.
Best Answer
If $\beta$ would be algebraic over $K$, then $\alpha = f(\beta) \in K(\beta)$ in contradiction with $\alpha$ transcendental over $K$.
Note: remember that any element of a simple extension $K(\gamma)$ of finite degree is algebraic over $K$.