Why is 3 not a prime in Q(sqrt7) despite that there is no integer in Q(sqrt7) of norm 3?
Like, if 3 can be expessed as a*b where a and b are non-unit integers in Q(sqrt7), then |N(a)|>=2, |N(b)|>=2 (they are not units), and |N(a)||N(b)| = N(3) = 9. Thus it must be that |N(a)|=|N(b)|=3. Contradictions, contradictions.. please help.
Also, what exactly are the prime factors of 3 (in Q(sqrt7))?