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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.

Alex A.G.
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  • Changed my question a bit to make it about only one thing, and that link did orient me in the right direction but couldn't quite finalise it. – Alex A.G. Feb 22 '24 at 17:37

1 Answers1

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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.

GreginGre
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  • I think you can say that $x^{q-1}=1$, but I still don't know how Lagrange helps. I've been thinking that $\mathbb{K}$ is the extension field of degree $n$ for a certain field $\mathbb{F}_p$ and with Lagrange, $p|p^n$ but I don't know how that helps :( – Alex A.G. Feb 22 '24 at 18:25
  • What gives Lagrange with the group $K^\times$ ? – GreginGre Feb 22 '24 at 18:31