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[Flaviano Bada]

I am new to studying abstract algebra (and math in general). I've been reading Gilligan and Pinter's books. I am trying to improve my understanding by doing exercises. However none of the books I am reading seem to come with exercise solutions.

Is there an abstract algebra book with lots of exercise with solutions?
I am confused as to why none of the math books come with complete exercise solutions. How do people verify that their answers are right.

I have read contemporary abstract algebra by Gallian. Though I dont remember if it had exercises and solutions but the chapters of the book are written in a very easy to understand way. It gave me a good insight into group theory.

I started to learn abstract algebra via YouTube playlists and am loving it so far. However, I am really struggling to find good exercises with answers that show the process to find the answer and don't just spoon-feed the answer. I tried finding books but many had answers to their exercices in other books that I couldn't find.

For graduate-level math books, the answer is typically not a value but a complete proof---typically of a related but relatively uninteresting topic. For example, one of the first exercises in the Neukirch book you reference is:

By the time that students are taking graduate-level mathematical courses, they are expected to have already mastered the general skills of constructing proofs. Seeing how somebody else has proved a point is thus not expected to be particularly educational, whereas struggling to prove something oneself forces a student to engage deeply with the material at hand.

Finally, examples of working with the concepts in the exercises are typically already given in the chapter, in the proofs of the main results, so adding extra examples by working proofs for the exercises would typically be of only incremental benefit, but undermine the value of students having to work through the proofs themselves.

Writing a graduate math textbook is a large effort that usually takes several years. The author typically has a vision of what material he/she wants to cover. After writing all the chapters and polishing everything, the exercises are probably the last part he/she works on. They are often meant as a pointer to additional more advanced topics in the literature that expand on the main content of the chapter, and adding solutions could require an effort comparable to writing an entirely new chapter (or several) to present that material in a polished, readable form. So, by that point the author feels that he/she is ready to move on to new projects and in any case the community is best served by releasing the book without exercise solutions. Solutions are sometimes added in later editions if the book is successful and the author is still passionate about the project.

Edit: Another thought that occurs to me is that adding exercise solutions can substantially increase the book's size. If the book is already of a good length (say 300 pages or more) then doing this could make the publisher very unhappy, and could potentially make the book less appealing to readers, who would start being intimidated/turned off by the book's length.

It is widely believed that there should be no solutions available, even privately, since this somehow ruins the game. This presumes that there should be "exercises" of the traditional sort in advanced mathematics courses, which is already partly dubious, since (as is often visible in commercially successful texts) it leads to make-work exercises often of questionable interest. I'd agree that there do exist significant, meaningful questions that may not fit into a small book... but would argue that then good write-ups of their solutions/resolutions should be available somewhere as models. Otherwise, all the students ever see is their peers' solutions... which in principle could be fine, but, observably, in practice, often overlook (through misunderstanding) ideas (from the text or otherwise) that make the resolution far more graceful and persuasive. That is, without good solutions, the only models anyone ever sees are "iffy".

(E.g., my abstract algebra text originally aimed to work a large fraction of the traditional significant questions as "examples", exactly to overcome the inertia of traditional-not-so-good alleged solutions of them, and have no "exercises" whatsoever. However, the publisher, who'd already made surprising concessions about intellectual property stuff, really-really wanted "exercises". So I made some near-clones of the worked exercises... And I've received several comments that I'm an anarchist for making those good solutions public!)

So, indeed, I think it's a bad idea to try somehow to suppress "good solutions". People will still grasp at bad solutions, and will be learning deficient versions of things to the extent they learn anything.

By the way, it is certainly not the case that the standard graduate mathematics texts provide means to resolve all their exercises. Often there is a considerable disjunction. Typically, the disjunction is that the theorems in the chapters do not at all suggest any quasi-algorithmic devices for doing computations in any particular case. E.g., abstract Galois theory usually disregards Lagrange resolvents, so does not hint at how to solve equations even when they can be proven solvable by radicals...

Nor is it the case that beginning math grad students are adepts at writing... so there is considerable feedback among them of marginal write-up style, marginal technical viewpoint, too much attention to secondary and tertiary details (often strictly demanded by in-my-opinion misguided texts or instructors), and needlessly distorted ambient language. Good writing models would help people "get over" this.

It's a commercial decision driven by the wishes of professors (who assign textbooks). Not having the answers makes them more important. I know of one author who wrote a very well regarded textbook with all the answers and had to remove them in second edition because his publisher said it wasn't selling as well.

I recommend people consult Schaum's, Kahn Academy, or look for books like Stroud or Granville that contain all the answers. It very much helps self study or even directed study (since the major learning comes NOT from the professor, but from working problems on your own).

Per the title, what are some of the oldest abstract algebra books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am already aware of the books of Dickson and van der Waerden.

The purpose of this book is to provide solutions and explanations for the exercises found in the third edition of Thomas Hungerford's "Abstract Algebra" textbook. It is intended to aid students in their understanding and mastery of abstract algebra concepts and techniques.

This book is primarily aimed at advanced undergraduate and graduate students who are studying abstract algebra. However, it can also be a useful resource for anyone interested in learning more about the subject.

Unlike many other solution manuals, this book contains detailed and rigorous explanations for each exercise, rather than just providing answers. It also includes extra exercises and examples to further enhance the reader's understanding of the material.

While this book is meant to supplement Hungerford's "Abstract Algebra" textbook, it can also be used as a standalone resource for those looking for extra practice and explanations in abstract algebra.

The author of this book is Dr. David S. Dummit, a professor of mathematics at the University of Vermont. He has also co-authored several other mathematics textbooks and has years of experience teaching abstract algebra.

For over six years now, I've been studying mathematics on my own in my spare time - working my way through books, exercises, and online courses. In this post I'll share what books and resources I've worked through and recommend and also tips for anyone who wants to go on a similar adventure.

Self-studying mathematics is hard - it's an emotional journey as much as an intellectual one and it's the kind of journey I imagine many people start but then drop off after a few months. So I also share (at the end) the practices and mindset that have for me allowed this hobby to continue through the inevitable ups and downs of life (raising two young boys, working at a startup, and moving states!)

I used to love mathematics. Though I ended up getting an engineering degree and my career is in software development, I had initially wanted to study maths at university. But the reality is, that's a very tough road to take in life - the academic world is, generally speaking, a quite tortuous path with low pay, long hours, and rife with burnout. So I took the more pragmatic path and as the years went by never really found the time to reconnect with math. That was until about six years ago when I came across Robert Ghrist's online course Calculus: Single Variable (at the time I took it, it was just a Coursera course but now it's freely available on YouTube). Roughly 12 weeks and many filled notebooks later, I had reignited my interest in math and felt energized and excited.

Growing up I always loved puzzles and problem solving. I would spend hours working my way through puzzle books, solving riddles, and generally latching on to anything that gives you that little dopamine hit.

If you're similar, mathematics might just be for you. Mathematics is hard. Seriously hard. And then suddenly, what was hard is easy, trivial, and you continue your ascent on to the next hard problem. It deeply rewards patience, persistence, and creativity and is a highly engaging activity - it's just you quietly working away, breaking down seemingly impossible problems and making them possible. I can't say enough how deeply satisfying and personally enriching it is to make the impossible, possible through your own hard work and ingenuity.

One thing many people don't know as well is that the mathematics you learn at most high schools is actually quite different from what you're exposed to at the university level. The focus turns from being about rote computation to logic, deduction, and reasoning. A great quote I read once is that for most of us, when we learn mathematics at school, we learn how to play a couple of notes on a piano. But at university, we learn how to write and play music.

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