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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?

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As far as I know, it does require knowledge of Galois theory to prove that quintic equations are not solvable. – Tunococ Sep 16 '13 at 9:17
I understand that that may be the case. I'm asking in general. I want "lower level" problems for incremental pieces of theory. – mehhhhh Sep 16 '13 at 9:17
I am curious as to what you consider "lower level". If I understand correctly, you might be feeling that beginning abstract algebra classes are usually not very motivating. I feel that too. Authors of algebra books usually feel that classification results of groups are practical enough applications. I did not share such appreciation at first, but now I do. Still, the topic doesn't interest me as much as commutative algebra and its application in topology and algebraic geometry. (You can tell I'm an Eisenbud fan.) – Tunococ Sep 16 '13 at 9:26
I would ask: why was the theory developed? If it was done just out of somebody's personal interest/curiousity, then maybe I won't be satisfied. If it was motivated by something, and it was successfully applied to that something, I want to know about it. This is a personal thing though. – mehhhhh Sep 16 '13 at 9:30
I think Wikipedia has some good information on the history of abstract algebra. According to Wikipedia, Number theory seems to be the first thing that motivated the development of abstract algebra. It was actually about commutative rings, not general groups. The interest in non-abelian groups stemmed the study of permutations, which was needed as roots of polynomial equations were studied. – Tunococ Sep 16 '13 at 9:40

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:

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M. M. Postnikov, Foundations of Galois Theory.

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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.

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