This is the first video in what I hope will be a series of solving old qualifying exam problems. In this video we solve Problem #9 from UC Irvine's Spring 2019 qualifying exam in algebra. In a previous video we proved that the pth cyclotomic polynomial (p prime) is irreducible over the rational numbers. I picked this problem as it uses the previous video as a prerequisite in a somewhat interesting way. I do not go into any details about rational canonical forms and take them as prerequisite knowledge. I'd be happy to make a video about this topic, should it be requested in the comments below!