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2026-04-17 13:35:07.1776432907
If $ [F(a):F]$ is prime, prove that $F(a)=F(a^n)$ for all positive integers n such that $a^n$ isn't in $F$.
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Suppose that $a^n$ not in $F$, you have $[F(a):F]=[F(a):F(a^n)][F(a^n):F]=p$, since $[F(a^n):F]\neq 1$ and divides $p$, $[F(a^n):F]=p$ since $p$ is prime. We deduce $[F(a):F(a^n)]=1$ and henceforth, $F(a)=F(a^n)$.