Let $F$ be a field of characteristic $p > 0.$ The claim in a certain lecture notes I am reading is that $F^{\star}$ (the multiplicative group of $F$) has no nontrivial elements of prime-power order. I am struggling to see why. Can you help?
2026-05-16 20:12:35.1778962355
About the multiplicative group of a field of prime characteristic
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I assume that powers of $p$ are meant, since otherwise it's clearly wrong.
Hint. The Frobenius map $x \mapsto x^p$ is injective since it's a field homomorphism.