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### In a finite field $GF(p^n)$, is it true that $a^{1 + p + p^2 + ... + p^{n-1}}$ always belongs to $GF(p)$?

In $GF(p^n)$ you can see $GF(p)$ as the set of all solutions of the equation $x^{p-1}=1$, together with zero. Thus, if $x=a^{1+p+\ldots+p^{n-1}}=0$ we are done. If not, then $a\ne 0$ itself. In that ...
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### In a finite field $GF(p^n)$, is it true that $a^{1 + p + p^2 + ... + p^{n-1}}$ always belongs to $GF(p)$?
Let $a \in \mathbb{F}_{p^{n}}$ be arbitrary and nonzero, and let $b = a^{1 + p + \cdots + p^{n-1}}$. We wish to show that $b$ is in $\mathbb{F}_{p}$. Note that \$1 + p + \cdots + p^{n-1} = \frac{p^{n}-...