In this note I will give a Galois-theoretic proof that for a prime and positive integer ,

I’d love to see a more elementary proof if you can come up with one.

First we need the following:

**Lemma 1** * Let be the cyclic group with elements. Let be a positive divisor of , and consider as a subgroup of . Then the number of automorphisms of which fix pointwise is equal to (which, in particular, is an integer). *

**Proof of the Lemma:** Note that any automorphism of fixes , though not necessarily pointwise: indeed has a unique subgroup of order , and thus any automorphism of must take this subgroup to itself. Thus we have a group homomorphism which is easily seen to be surjective; its kernel is precisely the subgroup consisting of those automorphisms of which fix pointwise. The statement follows by comparing orders.

Now to prove the initial claim, consider the field extension . Basic Galois theory tells that this is a Galois extension of degree . Consider the canonical homomorphism

which restricts an -automorphism to the group of units of . Clearly it is an injective homomorphism since is completely determined by where it sends the units. Moreover for any , lies in the subgroup of of those automorphisms fixing pointwise the cyclic subgroup of order , because the Galois group consists of -homomorphisms. By the lemma the subgroup of these automorphisms has order , whereas has order . This does it.

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Dear Bruno: Doesn’t it work if p is a power of a prime?

Dear Pierre-Yves: indeed, the same proof works! In fact it appears to work for any positive integer, but some other proof is needed for the general case.

Hi; I only just saw this. For any positive integer , couldn’t you just argue as follows? In the group of automorphisms of that fix multiples of pointwise (which is of order by your lemma 1), the automorphism given by multiplication by generates a subgroup of order , so divides by Lagrange’s theorem.

Nice! Thanks a lot, Todd. And thanks for reading!

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Neat! You’re just stocking your war chest of obnoxious problems to give as extra exam credit to any future abstract algebra students :)