- #1

mathmari

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The extension $\mathbb{Z}_p \leq \mathbb{F}_{p^n}$ is normal, as the splitting field of the polynomial $f(x)=x^{p^n}-x$ ($\mathbb{Z}_p$ is a perfect field therefore each polynomial is separable).

So, if $a \in \mathbb{F}_{p^n}$, then $q(x)=Irr(a,\mathbb{Z}_p)$ can be splitted over $\mathbb{F}_{p^n}$ (since all the roots are in $

\mathbb{F}_{p^n}$).

How can I find the splitting of $q(x)$ as an expression of powers of $a$?? (Wondering)