Hello All,

I had a few quick Quadratic Field related problems I wanted to throw out there to see if someone can give me a hand with.

1 - Does anyone know of a Quadratic Integer in Q[Sqrt(-1)] that is **prime** but whose __norm__ is not prime?

If by "quadratic integer" you meant "Gaussian integer" or just "integer", then the only primes in \(\displaystyle \mathbb{Z}\)* are elements of the form \(\displaystyle a+bi\) , with \(\displaystyle a^2+b^2=p\,,\,\,p\) an ordinary prime in \(\displaystyle \mathbb{Z}\) , or elements of the form \(\displaystyle a\,\,or\,\,ai\) , with \(\displaystyle |a|=3\!\!\pmod 4\) a rational prime, and the prime \(\displaystyle 1+i\).*

It's easy to check all these primes have a prime norm, so...

2 - What is a prime factorization of 6 in Q[Sqrt(-1)] ? I have been having trouble understanding these.

\(\displaystyle 6=2\cdot 3 = -i(1+i)^2\cdot 3=-3i(1+i)^2\) , with \(\displaystyle \pm i\) a unit, of course.

Tonio

Thank you! -Samson