# Thread: Integer Proof

1. ## Integer Proof

Let w be in Z(square root of -3) with w not equal to 0,1,-1. Prove w can be factored into irreducible elements of Z(square root of -3); that is, we can find irriducible elements p1,p2,...,pt with w=p1p2...pt

2. Originally Posted by teramaries
Let w be in Z(square root of -3) with w not equal to 0,1,-1. Prove w can be factored into irreducible elements of Z(square root of -3); that is, we can find irreducible elements p1,p2,...,pt with w=p1p2...pt
The factorization is not unique - you have:
$(1+\sqrt3i)(1-\sqrt3i)=4=2.2$.

But I think this can be shown by induction on |w| (which is a positive integer). Simply by the argument, that it is either irreducible, or it can be written as a product of two elements with smaller absolute value.

Of course, you'll have to show that all units in this ring have absolute value 1 first, but this is easy.