Decomposition of $z^q-z$ with $q=p^n$ in $\mathbb{K}[x]$ with $|\mathbb{K}|=q$.

Question

Let $q=p^n$ be a prime power, and $\mathbb{K}$ a field of $q$ elements. We need to show that $z^q-z$ decomposes into linear polynomials in $\mathbb{K}[z]$. Have been reading notes all over and suspect I could use some stuff about Lagrange, but don't really know what to do with it. I know $z^q-z$ is the product of monic irreducible polynomials, but don't know how to show these are linear.

Answer 1

Hints.

• If $x\in K$ is not zero, what can you say about $x^{q-1}$ (Subhint: Lagrange) ?

• Deduce that every element of $K$ is a root of $P=z^q-z$

• Conclude.