# Math Help - UFD Questions

1. ## UFD Questions

Hello all,

I have to questions regarding UFD's.

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

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

2. How can we prove that $Z$ 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.