A problem from G.E. Andrews's "Number Theory":

Suppose g is a primitive root modulo p (a prime) and m | p-1, where (0<m<p-1). How many integral solutions are there of the congruence

x^m - g "congruent to" 0 (mod p).

Proposed solution:

Assume there is at least one such x.

p ~|g since g is a primitive root.

Therefore p ~| x ^ m and p ~| x.

p ~| x implies x ^ p-1 "congruent to" 1 (Euler's Theorem).

mk= p-1 for some k. (hyp)

x ^ mk "congruent to" 1

x ^ m "congruent to" g. (hyp)

x ^ mk "congruent to" g ^ k. (raise to a power)

g ^ k "congruent to" 1. (Congruence is an equivalence rel.)

k < p-1

g is not a primitive root, since g does not belong to p-1.

Conclusion: there is no x such that x^m "congruent to" g.

Is this conclusion acceptable (or almost acceptable!) ?