Here's an easy question: What is the minimal polynomial of (1/3)+(1/6)(1+sqrt(-3))((1/2)(25-3sqrt(69)))^(1/3) + (1-sqrt(-3))/(3*2^(2/3)(25-3sqrt(69))^(1/3)
(insert this into Mathematica or your favorite program to make sense of it. If there's something missing, it's a root of x^5+x+1)
The minimal polynomial is, according to Mathematica, x^3-x^2+1, question is how to come to that conclusion? I've only seen this done with simple examples.