Solve the congruence mod in the ring .
Let . We claim that is prime. To that end, we claim that . Indeed, . Next we claim that and are proper. If to the contrary , then for some integers , , , and . Comparing coefficients mod 7 we have , a contradiction. Similarly, is proper. Now , so that . By Corollary 9.15, is prime.
Now we claim that . Indeed, note that mod . As we saw in a previous exercise, since is prime, mod . By Fermat’s Theorem, mod . In particular, mod . So we have mod . Reducing coefficients mod 7, we have mod , and so mod .