Suppose is a unique factorization domain, and is an integer in so that and have no common factor, but (norm of alpha) is a perfect square in . Show that is a perfect square in the quadratic integers in .
I'm not sure if I'm approaching this correctly but if and have no common factor, then and where and are prime in . But where . Somehow I need to show that where is a quadratic integer in (i.e., is a perfect square in the quadratic integers in ).