There are m distinct pᵏmth roots of unity over a field of characteristic p, for p a prime not dividing m

Let n = p^km, where p is a prime not dividing m, and let F be a field of characteristic p. Prove that there are m distinct nth roots of unity over F.


Note that x^n-1 = x^{p^km}-1 = (x^m)^{p^k}-1 = (x^m-1)^{p^k}. That is, the distinct nth roots of unity over F are precisely the distinct roots of x^m-1 over F.

If m=1, then certainly there is only 1 root of x-1.

Suppose m > 1. Now D(x^m-1) = mx^{m-1}, which has only the root 0 with multiplicity m-1. Clearly 0 is not a root of x^m-1, so that x^m-1 and its derivative are relatively prime, and thus x^m-1 is separable. Hence there are m distinct mth roots of unity over F, and so m distinct nth roots of unity over F.

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