Tag Archives: unique factorization domains

Unique Factorization Domains

Lets consider the number 2. Looking for a factorization of 2 over $ latex \mathbb{Z}$ we have 2 \cdot 1. Is this factorizartion unique ? If we are looking over another domain like $latex \mathbb{Z}[\sqrt{-6}]$ , then how easy is to find all the posible factorizations of 2 ? If we find one is it indeed unique?

2=(a+b\sqrt{-6})(c+d\sqrt{-6}) ,a,b,c,d \in \mathbb{Z}  considering the norm of 2 : N(2)=N((a+b\sqrt{-6}))(c+d\sqrt{-6}). But N(2)=(2+0\sqrt{-6})(2-0\sqrt{-6})=4.Taking norms for both sides we have :4=(a^2+6b^2)(c^2+6d^2) in $latex \mathbb{Z}$. On the RHS the factors can be a) either both of them 2 ,b) one of them 4 and the other 1.

a) It’s easy to so that the equation 2=x^2+6y^2 has no integer solutions (just consider (x,y)mod3).

b) 1=x^2+6y^2 \rightarrow 1=1+06 , which gives the factorization of norms: 4=4\cdot1 which is trivial factorization on $latex \mathbb{Z\sqrt{-6}}$ (we are looking for non-trivial ones).

One can read a little more diophantine_eq and how the integer equation 2x^3=y^2+1 can be solved over a unique factorization domain (UFD) such as  \mathbb{Z[i]}.

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