Originally Posted by

**Samson** Hello all,

I have to questions regarding UFD's.

1. Assume that Q[Sqrt(d)] is a UFD and that $\displaystyle a$ is an integer in Q[Sqrt(d)] where $\displaystyle a$ and $\displaystyle a$ (with a bar above it) have no common factor, but the norm of $\displaystyle a$ , N[$\displaystyle a$] is a perfect square in $\displaystyle Z$. Can someone show that $\displaystyle a$ is a perfect square in the quadratic integers in Q[Sqrt(d)] ?

2. How can we prove that $\displaystyle Z$ is a UFD?

Thank you all, I appreciate the help!