Introduction to abstract algebra

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1. postitJan 2019
   You first need to define your boundaries and limits. Don't get me wrong, but mathematics is grinding, and if you think you have an idea where you're getting yourself into because you've studied calculus and algebra as an undergrad CS, you will fail miserably. Don't make the same mistake I did :)

   Undergrad math is tooling; I like to use the analogy that it's like entering a workshop where you have all the tools available, but you're blindfolded, and you have no idea what each stuff is used for. So you'll have to touch everything. Look down for the tooling advice.

   When venturing in Mathematics, for your own sanity, please have an objective in mind. I'm down serious! Understand what you want to research and go in that direction.

   One last piece, Find a mentor, share, talk to people. You won't advance from yourself.

   As a piece of tooling advice, I have the following I've stolen from Reddit a few years ago (sorry I haven't found the source to it) __ TOOLING __ First of all, most important, GO LEARN ALGEBRA. Seriously, I know you think its bullshit but its the most basic skill in some ways that any mathematician should know. Second learn Calculus: Single and Multivariable. If you are still interested here are some things to go onto next:

   Discrete mathematics: This includes equivalence relations (probably one of the most important things for you understand ever), propositional calculus (logic) proof techniques (induction) and some basic combinatorics (Pigeonhole principle). You can literally find any textbook and start reading. The theory is kinda a hodgepodge, but those are the major themes.

   Linear Algebra: Again, one of the most important subjects you will ever study. Once you understand this, you are really on your way, and this stuff comes up everywhere. Many mathematicians have said many of the biggest proofs in the world come down to "just some linear algebra". The major point here is to understand that there is only one vector space for each dimension over a field and understand how a linear transformation becomes a matrix only after a choice of basis. Here equivalence relations come up again!

   Differential equations: Unless you're focused on engineering math or serious applied stuff, don't worry too much about this. Seriously, it's not that integral (haha get it!).

   Complex Analysis: Yes, mathematicians and Engr. Actually, do study "imaginary" numbers, but there is nothing imaginary here. This is serious stuff, do it.

   Okay, so now you're about as a sophomore/junior level place in mathematics. How to finish it off? It's not that unclear:

   Abstract Algebra -- Grab any book read about groups, rings, fields, vector spaces, and modules. Proofs will be difficult here but work through it. There are so many books here, avoid Lang (good book but not for starting out), Dummit/Foote is okay. As a undergrad I had a good time with Rotman's An introduction to abstract algebra.

   Analysis -- Grab Baby Rudin. No seriously, Grab this book, sit in a room for a semester and just fuckn' read it. You will basically be "redoing" calculus. This is a trial by fire, go!

   Topology -- Grab Introduction to Topology by J. Munkries. Its so well written it might as well be a coffee table book.

   There now, you have done everything a math major would. Yes, there are lots of things that are missing, arguably the most important things depending on what your goals are. Typically one studies Number Theory along with Abstract Algebra, or studies Analysis and Differential Equations together or Analysis and Topology. Seeing the links across different topics is essential, but I'm just giving the overview here.

   Not every mathematician studies logic, and there are LOT of fringe topics that I'm omitting (including some of my favs: Projective Geometry, Varieties, Lattice/Order theory, Combinatorics, Elliptic Curves, Coding theory, Harmonic Analysis, etc.). However, none of these are required courses at more than a 1% of programs

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2. empath75Dec 2018
   This is a fantastic introduction to abstract algebra. Highly recommended.
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3. qayxcSep 2022
   You know what, just for shits and giggles I actually DID open an introductory textbook on abstract algebra, just in case my undergraduate degree in maths failed me. Here's what the first chapter says about addition ('+') in Z:

     Here are 4 elementary properties that + satisfies:
     • (Associativity): a + (b + c) = (a + b) + c ∀a, b, c ∈ Z
     • (Existence of additive identity) a + 0 = 0 + a = a ∀a ∈ Z.
     • (Existence of additive inverses) a + (−a) = (−a) + a = 0 ∀a ∈ Z
     • (Commutativity) a + b = b + a ∀a, b ∈ Z.
   

   Also:

     Key Observation: There are naturally occuring sets (other than Z and Q) which 
     come equipped with a concept of + and ×, whose most basic properties are the 
     same as those of the usual addition and multiplication on Z or Q.
   

   I could go on with rings, fields, and vector spaces which also rely on the concept of addition as defined in Z, but I'm really curious to learn about addition being commonly used as a non-commutative operation, especially in the presence of an accompanying multiplication operation.

   edit: I forgot to provide proper references, so here's another example including a reference:

     When dealing with Abelian groups, it is customary to use additive notation.
     That is, the group operation will be called "addition," and 
     we will "add" two elements rather than "multiply" them. We write g + h instead of g • h or gh, and ng replaces g^n.
   

   [Walker, Elbert A., "Introduction to Abstract Algebra", 1987; Ch. 2, Pg. 70]
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4. graycatJul 2020
   Linear algebra, and more than one such book.

   IMHO long and still the best linear algebra book is

   Halmos, Finite Dimensional Vector Spaces (FDVS).

   It was written in 1942 when Halmos was an "assistant" to John von Neumann at the Institute for Advanced Study. It is intended to be finite dimensional vector spaces but done with the techniques of Hilbert space. The central result in the book, according to Halmos, is the spectral decomposition. One result at a time, the quality of von Neumann comes through. Commonly physicists have been given that book as their introduction to Hilbert space for quantum mechanics.

   But FDVS is a little too much for a first book on linear algebra, or maybe even a second book, should be maybe a third one.

   Also high quality is Nering, Linear Algebra and Matrix Theory. Again, the quality comes through: Nering was a student of Artin at Princeton. There Nering does most of linear algebra on just finite fields, not just the real and complex fields; finite fields in linear algebra are important in error correcting codes. So, that finite field work is a good introduction to abstract algebra.

   For a first book on linear algebra, I'd recommend something easy. The one I used was

   Murdoch, Linear Algebra for Undergraduates.

   It's still okay if can find it.

   For a first book, likely the one by Strang at MIT is good. Just use it as a first book and don't take it too seriously since are going to cover all of it and more again later.

   I can recommend the beginning sections on vector spaces, convexity, and the inverse and implicit function theorems in

   Fleming, Functions of Several Variables

   Fleming was long at the Brown University Division of Applied Math. The later chapters are on measure theory, the Lebesgue integral, and the exterior algebra of differential forms, and there are better treatments.

   Also there is now

   Stephen Boyd and Lieven Vandenberghe, Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares

   at

   http://vmls-book.stanford.edu/vmls.pdf

   Since the book is new, I've only looked through it -- it looks like a good selection and arrangement of topics. And Boyd is good, wrote a terrific book, maybe, IMHO likely, the best in the world, on convexity, which is in a sense is half of linearity.

   Some course slides are available at

   http://vmls-book.stanford.edu/

   For reference for more, have a copy of

   Richard Bellman, Introduction to Matrix Analysis: Second Edition.

   Bellman was famous for dynamic programming.

   For computations in linear algebra, consider

   George E. Forsythe and Cleve B. Moler, Computer Solution of Linear Algebraic Systems

   although now the Linpack materials might be a better starting point for numerical linear algebra. Numerical linear algebra is now a well developed specialized field, and the Linpack materials might be a good start on the best of the field. Such linear algebra is apparently the main yardstick in evaluating the highly parallel supercomputers.

   After linear algebra go through

   Rudin, Principles of Mathematical Analysis, Third Edition.

   He does the Riemann integral very carefully, Fourier series, vector analysis via exterior algebra, and has the inverse and implicit function theorems (key to differential geometry, e.g., for relativity theory) as exercises.

   All of this material is to get to the main goals of measure theory, the Lebesgue integral, Fourier theory, Hilbert space and Banach space as in, say, the first, real (not complex) half of

   Rudin, Real and Complex Analysis

   But for that I would start with

   Royden, Real Analysis

   sweetheart writing on that math.

   Depending on the math department, those books might be enough to pass the Ph.D. qualifying exam in Analysis. It was for me: From those books I did the best in the class on that exam.

   Moreover, from independent study of Halmos, Nering, Fleming, Forsythe, linearity in statistics, and some more, I totally blew away all the students in a challenging second (maybe intentionally flunk out), advanced course in linear algebra and, then, did the best in the class on the corresponding qualifying exam, that is, where that second course was my first formal course in linear algebra.

   Lesson: Just self study of those books can give a really good background in linear algebra and its role in the rest of pure and applied math.

   No joke, linear algebra, and the associated vector spaces, is one of the most important courses for more work in pure and applied math, engineering, and likely the future of computing.

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5. omar_a1Aug 2019
   Not sure that you'll find any "Made Easy"-type (abstract) algebra texts, but my college textbook was "Introduction to Abstract Algebra", by W. Keith Nicholson, for what it's worth.
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6. sedachvJan 2015
   W. Keith Nicholson's Introduction to Abstract Algebra covers a lot of different topics in mathematics from an algebraic perspective.
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