## Solve a linear congruence over a quotient of an algebraic integer ring

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 .

Indeed, .

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