My question is from this note or the answers of the following questions.
Perfect field of characteristic $p$ Equivalent definitions of perfect field
I have a question about the the proof of the above. The two are both about proving that a field $F$ whose characteristic is p(prime) is perfect iff $F^{p}=F$.
They proceed as follows: If $F^{p}\neq F$, pick $a \in F - F^{p}$. Then $x^p-a$ is irreducible and inseparable in F[x]. My question lies in the process to show that $x^p-a$ is irreducible. Any nontrivial proper monic factor of $x^p-a$ is $(x-\alpha)^{m}$ where $\alpha$ is a zero of $x^p-a$ and m is a natural number between 1 and p-1. The coefficient of $x^{m-1}$ is $-m\alpha$ and for $(x-\alpha)^{m}$ to be in F[x], $-m\alpha \in F$. I followed up here. But they said that m is invertible and conclude that $\alpha \in F$. Why is this true?
Here is my thought: since m is between 1 and p-1, we can naturally consider m as an element in the prime field of F, so is an unit. Am I right?
Yes, of course, I don't see why you would have any doubts. Though I would rather rephrase it as: $m$ is an integer, so it can be seen as an element of the prime field, and since it is between $1$ and $p-1$ it is not divisible by $p$, and therefore is not $0$.