What is wrong in my reasoning on field extensions?

Question

In my exercise, K is a field of characteristic p, and $K \subset L$ is a finite extension. $ x \in L$ and I must prove that $K[x^p] = K[x]$ if and only if x's minimal polynomial is separable.

However I "managed to prove" that $K(x^p) = K(x)$ is always true , but I don't see where is the flaw.

My proof: $f(x^p) = f(x)^p$ for $f \in K[X]$ (characteristic p) so f(x) = 0 if and only if $f(x^p)=0$ and x and $x^p$ have the same minimal polynomials.

Since $K(x^p) \subset K(x)$ and since they have the same dimension (degree of their shared minimal polynomial) they are equal.

What is wrong? Thanks!

Answer 1

To elaborate on my comment, it's not generally true that $f(x)^p=f(x^p)$. Let $f(x)=\sum_i a_ix^i$. Then $$f(x)^p = \sum_i a_i^p x^{ip},$$ but $$f(x^p) = \sum_i a_i x^{ip}.$$ These are not in general equal, since $a_i^p\ne a_i$ unless $a_i$ is in the characteristic subfield.

This is the core error in your proof.