clarche olyana haydun
Kansas Eiffel
For his leadership in modern algebraic geometry, including three major bodies of work: tale cohomology; algebraic approximation of formal solutions of equations; and non-commutative algebraic geometry.
Last semester I picked up an algebra course at my university, which unfortunately was scheduled during my exams of my major (I'm a computer science major). So I had to self study the material, however, the self written syllabus was not self study friendly (good syllabus overall though).
So, now I want to ask whether any of you know any good books on abstract algebra, which lift off at basic ring theory and continue to more advanced ring theory and to finite fields, Galois theory, ...
There's always the classic Abstract Algebra by Dummit and Foote. Section II of the text gives a nice treatment of ring theory, certainly providing plenty of review for what you have already covered while introducing more advanced concepts of ring theory. Section III will cover the field and Galois theory you're interested in. Some of the exercises can be difficult at times, especially for self-study, but the authors tend to give a number of examples and always provide the motivation for why they are doing what they are doing.
"Contemporary Abstract Algebra 7/e provides a solid introduction to the traditional topics in abstract algebra while conveying to students that it is a contemporary subject used daily by working mathematicians, computer scientists, physicists, and chemists. The text includes numerous figures, tables, photographs, charts, biographies, computer exercises, and suggested readings giving the subject a current feel which makes the content interesting and relevant for students."
It's a unique book that covers the basics of group theory, ring theory, and even a tiny bit of Galois Theory, but it does it almost entirely through problems. Every chapter begins with a short section defining some terms and giving a few basic proofs, and then it leads the reader through the rest of the exposition in a series of problems, some difficult, some not. The end result is that if you actually do all the problems, you've written the book yourself. It's impossible not to be comfortable with basic abstract algebra if you take this book seriously.
One book that I did not see mentioned, but which really deserves some accolades is the recent book Visual Group Theory by Nathan Carter. There are some excellent accompanying videos by Prof. Macauley on his youtube channel. These go really well together.
The biggest trouble I ran into with group theory and abstract algebra was the dizzying set of definitions that most books present at the beginning. You get a bunch of definitions with little or no motivation and with little description of the underlying geometry of how the binary operations work. The nice thing about the Carter book and the videos is that it spends a lot of time working though group diagrams and showing the "symmetry" of a group. It is easy to get caught up in the formalism, but without a good intuitive understanding of how different groups work--and how simple groups differ--it is easy to get frustrated--especially in self study. Also note that the Carter book has exercise solutions at the end.
There is no easy or right answer. I know brilliant professors who cannot easily decide what textbook to use for an advanced math course, and for good reason. Every book has its own strengths and weaknesses. I suggest you go to your math library (assuming one is available during this pandemic) and examine several books. A book you like might be hated by someone else, it is highly individual. You likely will need at least two or three books so you can go back and forth. Even a good book can be bad in a particular section and vice versa. Use a common textbook that has gone through at last two or three editions as a guide as to what topics to cover and then be prepared to use alternate books to actually learn the topic.
Use the Internet. Don't be afraid to read lecture notes or check Wikipedia. Also, Professor Keith Conrad (Univ. of Conn.) has dozens of expository papers on algebra on his web site, some are easy, some are difficult, and some are advanced or specialized. I have found that lectures by professors at lessor known universities to often be better than those by professors at famous brand name universities. That being said, I have found lectures by Unv. of Berkeley professors to be quite good, and lectures by MIT professors to also be good, but the latter are often very fast paced and better for review than to self learn from as they are so intense.
I suggest you get an easy book, an intermediate book and eventually a hard book. Herstein's: Topics in Algebra is harder than Birkhoff and MacLane's book, but Birkhoff and MacLane's book is good for learning the fundamentals. As an undergraduate I used Herstein, but I think it is too difficult to self study from.
It is critical to learn the definitions and other fundamentals cold and then go on to a more advanced treatment. (One really smart professor basically told me: memorize definitions, but do not memorize proofs, just understand them.) Herstein loves to give problems and results that are hard using elementary methods, but easy using more advanced methods. In my opinion this a bad way to learn, as not everybody is clever at solving hard problems or following highly technical arguments, and I think it is more useful to put one's energy into learning the concepts and theory that makes it possible to eventually easily understand what is really going on, rather than rely on clever technical tricks or manipulations to get a result with no real deep understanding as to what is really going on. Neither Herstein nor Birkhoff and MacLane cover everything a graduate course would cover. Herstein, in my opinion, makes the subject seem more difficult than it is.
I also use: Algebra: A Graduate Course by I. Martin Isaacs, it has its strengths and weaknesses. It is is elegant and the proofs are carefully done, but it may be too abstract and condensed to self study from.
A classic is the two volume (mostly of the time only the first volume is used) set by B. L. van der Waerden titled: Modern Algebra. There is at least two English editions (the original is in German). Even though the editions differ, any English edition is fine. And if you really groove on abstraction there is Serge Lang's book, simply titled: Algebra.
Good luck. I think it is great you are so motivated. Keep the faith. Don't worry if at times you get overwhelmed or discouraged --- self study is not easy --- it has happened to many of us at some point in time, yet somehow we didn't let it stop us, nor should you.
I am speaking from the standpoint of a student, and I think that a very good book on introductory abstract algebra that doesn't get mentioned very often is Basic Algebra by Anthony W. Knapp. From experience, the text is accessible with very little pre-requisite knowledge, is less "talkative" than Dummit and Foote (and in my opinion, definitely not dry, unlike D&F), and more rigorous in exposition than Artin's Algebra, although Artin's book is a good and standard first text as well.
Knapp covers most basic topics that the undergraduate student needs to know and is largely self-contained. I think, for the first seven chapters of this book, you can't really do much better by way of alternative texts. However, you could supplement or even replace the eighth chapter with Introduction to Commutative Algebra by Michael Atiyah and Ian MacDonald. However, if you are reading algebra for the first time, I don't suggest using Atiyah's book, unless you are feeling very confident or very lucky! :) Having said that, it is an excellent book and you should try reading it at some point. For the ninth chapter, you could use Emil Artin's classic little book on Galois Theory, based on his lectures on the subject. Another good reference which I haven't used but heard quite a few good things about is Nathanson's Basic Algebra: I (Chapter 4 (?), I think). Yet another book on Galois Theory is D.J.H. Garling's Galois Theory, which is where I initially learnt my Galois Theory from. As for Chapter 10 in Knapp, I have nothing to say, since I never got down to reading it.
Fundamentals of Abstract Algebra by Malik, Sen & Mordeson is a very good book for self study.The topics are covered in detail with many interesting examples and exercises.Also it provides hints and answers to difficult questions making it suitable for self study.
I have utilised many books but among them "Fundamental Of Abstract Algebra" by Malik, Moderson, Sen is really good covered with every topic what an undergraduate student usually requires and provided different types of exercises. For initial understanding and concept making, it is really a good one. Full pdf version is available in online, you may try it. Thank you ?.
The formula $z=\frac-b+\sqrtb^2-4ac2a$ expresses the solutions to the quadratic equation $az^2+bz+c = 0$ in terms of the inverse of an analytic function $z \mapsto z^2$. We have simply turned the problem of inverting one analytic function, $z \mapsto az^2+bz$, into the problem of inverting another analytic function, $z \mapsto z^2$. Therefore, all the power of the quadratic equation lies in how it solves any quadratic equation by inverting a single analytic function, $z \mapsto z^2$.
Similarly, Cardano's formula solves any cubic equation by inverting a two analytic functions, $z \mapsto z^2$ and $z \mapsto z^3$. Interestingly, you can also solve a cubic by inverting only one analytic function, for example $z \mapsto \sin z$.
I know very little about the status of this question (exept that it holds for some small values of $n$). Any information on what is known about this question would also be of interest.
I think that a large part of the difficulty we have in understanding why this result is considered important is that it is psychologically difficult to put oneself into the shoes of mathematicians of the past. There was a time not so many centuries ago when people didn't know how to solve cubic equations with radicals. Whether the quintic is solvable in radicals was once a difficult question. A problem that gains some notoriety for being difficult is usually going to be considered important when it is finally solved, regardless of whether it ends up occupying a central place in the "theory" that we end up constructing a posteriori. Fermat's Last Theorem is another good example of this.
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