This morning at the Godsil 65 conference, Cheryl Praeger was wearing a T-shirt she was a bit ashamed of. It had been produced by the Australian Mathematics Trust to commemorate Niels Abel’s anniversary, and featured a quintic equation (he was the mathematician who first showed conclusively that the quintic is not soluble by radicals). Unfortunately the quintic they had chosen has an integer root!
So here is my suggestion for a problem which should be easy if you have done any Galois theory. The other thing Abel is remembered for is abelian groups, so why not an irreducible quintic with abelian Galois group?
Exercise: Find a simple example of such a quintic.