I have been going through.Algebraby I.M. Gelfand and A. Shen

"Problem 122" has been giving me difficulties. For those who have worked through this "classic" book, how does one even approach these types of problems?

My difficulties start with sub-problem "c." It wants the reader tofactor a.^{10}+ a^{5}+ 1

In high school you only learn a handful of factoring techniques. This book shows me that is not sufficient. Where do you start?

And if you can solve this problem, could you please walk me thorough? Thank you.