Hi, I'm having problems with the second part of this problem.

Let $\displaystyle F$ be an extension of $\displaystyle K$ and $\displaystyle a \in F$ transcendental over $\displaystyle K$.

Let $\displaystyle f \in K[x]$ with $\displaystyle \text{deg } f >0$. Show:

a) $\displaystyle f(a)$ is transcendental over $\displaystyle K$.

b) If $\displaystyle b \in F$ and $\displaystyle f(b)=a$, then $\displaystyle b$ is transcendental over $\displaystyle K$.

For part a, I assumed otherwise, so there exists a $\displaystyle g \in K[x]$ such that $\displaystyle g(f(a))=0$, but $\displaystyle g(f(x))=h(x) \in K[x]$ and $\displaystyle h(a)=0$ which contradicts $\displaystyle a$ being transcendental over $\displaystyle K$.

For part b, I'm not sure. I tried doing it by contradiction also, but I'm not getting anywhere. I feel like it's relatively simple, and I'm just forgetting something obvious. Any help would be appreciated! Thanks!