Do you know how to solve the below two questions? If you do , please teach me. Thank you very much.
Prove that Z is a UFD.
Suppose Q[sqrt(d)] is a UFD, and alpha is an integer in Q[sqrt(d)] so that alpha and alpha bar has no common factor, but N(alpha) is a perfect square in Z. Show that alpha is a perfect square in the quadratic integers in Q[sqrt(d)].