abstract algebra example book

Question

It's very exciting when you can use the theory to solve "lower level" problems. For example, I'm looking forward to understanding why the quintic equation is not solvable. In the undergraduate curriculum, that seems to happen very late: only if you decide to take a class in "galois theory".

My experience with my first abstract algebra class is that it just develops theory ("why can such a group not exist? what properties must this structure have?", etc). I want to be able to see how the theory helps us in the lower level sense. Are there "example books" with interesting "lower level" problems?

Answer 1

M. M. Postnikov, Foundations of Galois Theory.

Answer 2

I recently learned some basics of the theory of Gröbner basis. These mainly deal with a solution of polynomial equations. Along the way, an ideal naturally arises and Hilbert Basis Theorem plays an important role.

The book I read about this topic was Ideals, varieties, and algorithms by Cox, Little, and O'Shea. I think you'd like it.

Answer 3

I am not sure if it's Galois theory you are, at least somewhat specifically, interested in. If so, you can find links to what I consider to be outstanding notes with exercises as well as solutions by Andrew Baker:

http://www.maths.gla.ac.uk/~ajb/course-notes.html