## Factor the principal ideals in an algebraic integer ring which are generated by ramified rational primes

Let and let be the ring of integers in . Factor in , where is a ramified rational prime.

By Theorem 9.6, if does not divide the discriminant of (using Theorem 6.11), then is not ramified. Now , so that if is ramified in , it is either 2 or 5.

We claim that . Indeed, the direction is clear, and we have . We claim also that is maximal. To this end, let , and say mod 2 where . Evidently, mod ; if mod 2 then mod , and if mod 2 then mod . Now suppose ; then . Comparing coefficients mod 2, we have mod 2, a contradiction. So , and thus is a field. Hence is maximal, and is the prime factorization of .

Certianly . We claim that is prime. To see this, note that . If , then mod , where . Suppose . Then for some . Comparing coefficients, we have mod 5. In particular, mod , where are distinct. Thus is a field, so that is maximal. Thus is the prime factorization of .

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