1. ## UFD Questions

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!

2. 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!
I'm sure that there must be a generic prove to show that Z is a UFD. Here is what I have found thus far:
http://people.ucsc.edu/~smohare/UFD.pdf

Can someone turn that into Lehman's terms? Also, I still could use some help understanding part 1 as well.