Characterize the irreducible elements in .
Suppose is irreducible. By our proof of Theorem 8.5, it follows that for some unique positive rational prime . That is, for some . Now , so that either or . (Recall that is a rational integer and not 1.) Let .
Suppose . Then , so that is a rational prime.
Suppose . Now , so that (by Theorem 7.3) is a unit. (In Theorem 7.9 we saw that the units in have the form where .) Thus is an associate of in . We claim that in this case, there does not exist a solution of the equation . If so, then . But since is irreducible, either or is a unit- so , a contradiction.
In summary, if is irreducible in , then either (1) is a rational prime or (2) is the square of a rational prime and the equations have no solutions in .
Conversely, if is prime, then is irreducible. If where is a rational prime and if has no solution, then no element of has norm , so that is irreducible.