>I'm told that if all rings of the form a+b.sqrt(-p) are unique factorization domains then there is a fairly simple proof I just can't resist showing why this

>I'm told that if all rings of the form a+b.sqrt(-p) are unique factorization domains then there is a fairly simple proof I just can't resist showing why this is false. It's actually pretty simple. Take p=5. Then our 'integers' in this ring are of the form a + b\*sqrt(-5) for integers a, b. Taking b = 0 we see that all the normal integers are part of this ring. So consider how 6 factorizes. On the one hand we have 6 = 2 × 3... uncontroversial. But... due to the existence of new numbers in this expanded ring, we also get 6 = (1 + sqrt(-5)) × (1 - sqrt(-5)) . These are evidently two different factorizations... it suffices to show that both are prime factorizations under the modified definition of 'prime' we get when we allow factors of (a + b × sqrt(-5)). You can get there by noticing that any number of the form a + b × sqrt(-5) that factors into smaller such numbers must either be totally real (b=0), be totally imaginary (a=0), or have a larger absolute value than any number that could possibly multiply together with another number > 1 to get you 1 +/- sqrt(-5). This line of reasoning can show that 2, 3, (1 + sqrt(-5)), and (1 - sqrt(-5)) are all primes in this ring.