I've read the proof in my textbook, but it eventually boils down to contradiction, and in most cases the downside to proof by contradiction is that it removes some intuition. I also couldn't find any other explanation online! Can anyone give me an intuitive explanation as to why finite fields are of order $p^n$? I think this is a super cool fact, but I find it kind of hard to appreciate when I don't have the big picture in mind.
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1Usually it is used that the finite field is a vector space over the field of integers mod p. – coffeemath Dec 07 '23 at 04:33
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1I don't think you are correct in claiming it "boils down to a contradiction." Anyway... Duplicates: one, two, three, four. – Arturo Magidin Dec 07 '23 at 04:36
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Power residues form a group under multiplication is the closest my knowledge comes to this idea. – Roddy MacPhee Dec 07 '23 at 23:20