UFD Questions

Hello all,

I have to questions regarding UFD's.

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

2. How can we prove that is a UFD?

Thank you all, I appreciate the help!

Quote:

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Originally Posted by Samson

Hello all,

I have to questions regarding UFD's.

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

2. How can we prove that is a UFD?

Thank you all, I appreciate the help!

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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.