While exploring concepts related to field extensions, I came across the following statement:
"Let $K$ be an extension field of $F$ and $u\in K$ an algebraic element over $F$. Consider the homomorphism $F[x]\to K$ defined by evaluation of a polynomial at $u$. Since the image is a subring of a field, the kernel is a prime ideal in the PID $F[x]$"
How does one prove the final sentence?