Abstract Algebra Pdf Notes
Desiderato Chouinard
Warning up front: if you go into class tired or distracted, my suggestion is to write everything down that is written on the board, or even more if your presenter has a more oral style. The reason is that you (a) probably aren't very good at discerning what is important in your state, and (b) are very likely to fall asleep if you do nothing but listen.
Assuming that you are focused, the single most important thing happening in a math class is your inner monologue. You can't write it all down, of course; it goes too fast. But whenever I find myself saying something different than the presenter or asking a question, I write it immediately; even if I am already in the middle of writing something else.
In theory, I think that this is all one ever really needs for notes. In practice, some other things are useful. The gist of what follows is: your notes should contain approximately the difference between the entire class and what is in the book.
Examples: Many. Some will be standard enough that they are in the book, but oftentimes the examples are chosen specifically because they are really insightful, even if they don't appear so on the surface. I don't want to say "all" because sometimes a presenter will go overboard, but you want lots of them.
Definitions: Only nonobvious ones. There are lots of technical, obvious definitions. Don't bother with these. Also don't bother writing "for all natural numbers $n$", or somesuch, unless it is radically important that it is not, say, an integer, but this fact isn't obvious from the context.
Theorems: Really important ones, and anything starting with the phrase "The Following Are Equivalent". Usually a presenter will make a special effort to distinguish the groundbreaking proofs; I mean on the scale of Mean Value in analysis, Rank-Nullity in linear, or Lagrange's Theorem in abstract. You don't technically need these because they're in the book, but since these are the ones you should remember long after the class ends, some extra effort would be good.
Proofs: I take a pretty radical stand here. If your textbook is not Rudin-scale terse, I don't think it is necessary to write any proof given in a lecture*. You will be writing proofs in homework, so it's not about practice. Any proof given in class is either simple enough that you can do it yourself or significant enough that it will be given by the textbook; and even if not, you can find it online. Your attention is much better spent by concentrating very hard during the lecture and gaining an intuition about the argument. Try to keep an example running as you go through it**. Your goal is to get a very big picture of the proof.
Do not memorize proofs unless you have a very specific reason to do so (i.e. you will be asked to quote the proof of the first Sylow theorem on the midterm). Copying down proofs from a book may have some benefit for you, it doesn't for me. Be rigorous on the homework, and if your professor does not assign all the problems in the section, think about how you might solve the others.
[** actually, this is something I am experimenting with: I'm auditing an Algebraic Geometry course and I'm taking notes by phrasing all of my theorems as examples (e.g. instead of "A ring without zero divisors has the cancellation condition" I would write "$\Bbb Z[i]$ has the cancellation condition" and phrase the proof through that lens). I'm not sure if it is actually useful yet.]
Math is a delightfully introspective subject. I know more about my innermost thoughts and the deeper workings of my mind through hours staring at a whiteboard than a psychologist could gather in a hundred years.
From this, I know that I cannot take notes. Ever. If I have any form of paper in front of me during a lecture, then I have a blank slate for my thoughts and math, usually unrelated to whatever topic I'm trying to learn, will soon have pervaded the paper. Instead, I have to prepare extensively before class and simply listen to lectures I attend.
I include this as a helpful guide toward developing your study/lecture/learning habits. You, and only you, can figure out what works best. My suggestion is to experiment. Once you find a method that works well for you, stick with it.
Probably contrary to most of your academic experiences thus far, you cannot rote memorize math past this point and expect to do well. We cannot "just know things." For a related, anecdotal reference on the habit of just learning machinery, see this post.
You will only understand the theorems if you use them enough to know them by heart. Play with them! If you intend to become a mathematician, these will be your toys for the rest of your life. If you do not wholeheartedly want to work/play with these for the rest of your life, then you might want to consider changing your major.
Now, the reputation of this class precedes itself and it is very likely that you've heard about it from upperclassmen complaining about it. And you may be thinking to yourself: "Wait! I already took MATH 33A, which is Linear Algebra. Why do I have to take it again? In fact, MATH 33A is a pre-requisite for MATH 115A!"
In introductory courses like Math 33A, linear algebra often revolves around matrix studies. However, at the 115A level, the focus shifts to exploring vector spaces and their transformations. If you're unfamiliar with the concept of a vector space, don't worry - we'll delve into it soon. For starters, consider $\mathbbR^n$ - the set of all $n$-tuples of real numbers - as your introductory vector space. Typically, this is the sole vector space explored in elementary linear algebra courses. In 115A, however, we'll expand our horizons, exploring linear algebra in various other vector spaces, which proves to be incredibly beneficial.
Our approach involves starting from the very basics. It is perhaps helpful to momentarily set aside all your previous mathematical knowledge and treat 115A as a foundational course designed to systematically build a specific mathematical field from the ground up. This is our initial aim in 115A.
A noteworthy point regarding this goal is the following: You might be anticipating that exploring linear algebra in vector spaces beyond $R^n$ will be a radically different and exciting experience. However, I must clarify that abstract linear algebra in general vector spaces largely mirrors the linear algebra you've encountered in $\mathbbR^n$. The concepts of linear independence, transformations, kernels, images, eigenvectors, and diagonalization - all familiar topics within the realm of $\mathbbR^n$ - function similarly in 115A.
(2) Construct and Follow Abstract Mathematical Arguments and Statements
This goal extends beyond mere proof-writing. Upper-division mathematics, in contrast to lower-division studies, prioritizes the discovery and articulation of truths over computation. In 115A, every solution you formulate should be viewed as a mini technical essay, marking a departure from mere scratch work to determine problem solutions. Mastering the art of clear, logical, and effective communication of mathematical truths is a challenging yet essential skill to develop.
Here is a list of strategies and advice that I found useful in navigating this challenging yet rewarding course. Some of these advice are in retrospect (i.e. things I would do if I were to re-take the course).
3 years ago, I experimented with the 'weekly newsletter' approach and it was not only unsustainable, but also slightly spammy. My new model is as follows: I promise to only send an email to your inbox when I'm absolutely positive it is something you'll find interesting, and perhaps more importantly, actionable. Topics will include personal finance, productivity or general life insights.
Abstract algebra is the branch of mathematics that deals with abstract or generalized mathematical structures, such as groups, rings, fields, and vector spaces. It studies the properties of these structures and their operations, and how they relate to each other.
Abstract algebra has numerous applications in various fields, including cryptography, coding theory, physics, chemistry, and computer science. For example, group theory is used in particle physics to understand the fundamental interactions between particles, and coding theory uses algebraic structures to design efficient error-correcting codes.
Some popular books on abstract algebra include "Abstract Algebra" by David S. Dummit and Richard M. Foote, "A Book of Abstract Algebra" by Charles C. Pinter, and "Algebra" by Michael Artin. It is important to choose a book that suits your level of mathematical knowledge and learning style.
Like any branch of mathematics, abstract algebra can be challenging to learn, but it also has a logical and systematic structure that can make it easier for some people. With dedication and practice, anyone can grasp the concepts and techniques of abstract algebra.
To understand and appreciate abstract algebra, one should have a strong foundation in basic mathematical concepts such as sets, functions, and proofs. It is also helpful to have some knowledge of linear algebra, as it provides a good introduction to abstract structures.
An Introduction to Algebraic Combinatorics: Power series and generating functions, partitions, permutations, alternating sums and Schur polynomials. Work in progress, but mostly complete.
Enumerative Combinatorics: Rigorous and detailed introduction to enumerative combinatorics. Chapters 1 and 2 done, covering various types of subset counting, inclusion-exclusion, binomial identities and more. Further topics are covered in the Fall 2022 lecture notes.
Darij Grinberg, Notes on the combinatorial fundamentals of algebra (PRIMES 2015 reading project: problems and solutions).
Sourcecode.
A version without solutions,for spoilerless searching.
A set of notes on binomial coefficients, permutations anddeterminants. Currently covers some binomial coefficientidentities (the Vandermonde convolution and some of its variations),lengths and signs of permutations, and various elementary propertiesof determinants (defined by the Leibniz formula).
The sourcecode of the project is also trackedon github.