# Field is perfect iff every element has pth root.

I am trying to understand the proof for one direction of the following theorem:

I am confused about the part in red. Why does this work? Has it something to do with the Frobenius endomorphism?

• Exactly. That' because the Frobenius endomorphism is indeed an ring homomorphism. – Crostul Mar 18 at 20:20
• @Crostul So the Frobenius function can be applied to polynomials? – numericalorange Mar 18 at 20:25
• It can be applied in any commutative ring of characteristic $p$. In particular it can be applied to the ring of polynomials over a field with characteristic $p$. – Crostul Mar 18 at 20:31
• Wow, thanks so much for clearing my head! – numericalorange Mar 18 at 20:31