Questions tagged [abstract-algebra]

For questions about the study and teaching of abstract algebra, including topics such as groups, rings, fields, and vector spaces.

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An algebra student wants to learn to type commutative diagrams in LaTeX

A maths student takes his first course on homological algebra, and wants to write his answer in LaTeX. What is the simplest way for him to write commutative diagrams in LaTeX?
guest's user avatar
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4 votes
0 answers
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Learn ring theory more fun

I asked this question at 'mathoverflow'. However, the character of this question does not match mathoverflow, so I am asking this question again on this site. Motivation: Among abstract algebra, I am ...
pokssin's user avatar
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10 votes
5 answers
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Requesting a Polynomial System of Equations

I am teaching a course in commutative algebra, and it includes a project where the students research on a particular topic, solve a small problem and present it to the class. I usually give my ...
Aravind Gundakaram's user avatar
2 votes
2 answers
287 views

Introduction of group action as morphism of groups

The usual definition of a group action is as follows. Let $G$ be a group and $A$ be a set. An action of $G$ on $A$ is defined to be a map $\rho:G\times A\rightarrow A$ satisfying certain conditions. I ...
Praphulla Koushik's user avatar
1 vote
1 answer
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On Teaching Cyclic Groups [closed]

My professor asked me to take 20 mins on talking about Cyclic Groups and their practical applications in front of the whole class. How should I start? Can anyone please help me.
Florence's user avatar
4 votes
4 answers
698 views

Third isomorphism theorem: how important is it to state the relationship between subgroups?

In texts which present the third isomorphism theorem: $$(G/N)/(H/N) \cong G/H$$ the relationship between the entities is often seen presented in the form: Let $H$ and $N$ be normal subgroups of a ...
Prime Mover's user avatar
6 votes
3 answers
313 views

Visual aids for understanding group theory

I want ideas for pictorial representation of groups which can help one understand the different group theorems. Here are some examples of the type of thing I am looking for. In this video by socratica ...
tryst with freedom's user avatar
5 votes
3 answers
626 views

Should one teach to use equality or isomorphism in particular groups?

I am wondering the following. Suppose we have some particular space $X$ and $x_1,x_2,x_3,x_4\in X$ has the law of composition that works like Klein four-group. Is it correct to say that the structure ...
guest's user avatar
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3 votes
1 answer
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What is a good way to broach the subject of abstract algebra, to a student in Calc 1, or pre-calc?

Background I want to introduce my students to some big names in mathematics, and one of the names I want to bring up is Richard Borcherds, known for his contributions to the fields of number theory, ...
Alec's user avatar
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4 votes
1 answer
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is it beneficial to encourage high school students to conduct their own 'research' in mathematics?

Is it good idea to encourage students to look up more about particular topics that interest them? The idea is that I don't think they will understand how to read math literature. for example, if a ...
Lenny's user avatar
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1 vote
2 answers
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Advice on teaching advanced mathematics to high school students

I am a high school student really into algebra and algebraic geometry. I want to expose this sort of math to other high school students that have the motivation and ability. For a long time, I had no ...
John Clever's user avatar
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Can you talk about (the rest of the) field axioms when the operations are not closed? [closed]

Note: Updated based on this. In my course, my instructor posed the following exercise: Let $S$ be the subset of $\mathbb R^n$, $S=\{(a_1,a_2,a_3...a_n) | a_2 = \pm a_1, a_3=...=a_n=0 \}$. Define ...
BCLC's user avatar
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5 votes
2 answers
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Undergraduate-level abstract algebra books or courses that don't start with groups or rings

When I was an undergrad studying abstract algebra, we used the second edition of Artin and covered groups first and then rings. Fields, vector spaces, and algebras came later, I think. I remember ...
Greg Nisbet's user avatar
1 vote
1 answer
256 views

in what sense is the subject of finite group theory 'algebraic'?

[cross posted from mse] the class of all finite groups is not closed under produtcs - example: the product over all finite cyclic groups - thus it is not a variety of algebras, ie, it's not ...
ac15's user avatar
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7 votes
3 answers
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Rings in parallel with groups in abstract algebra

In a previous question, I asked about the pros and cons of teaching rings before groups in abstract algebra. Recently, it has come to my attention that there is a third approach - a unified approach - ...
J W's user avatar
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2 votes
1 answer
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Improving exposition of a proof about polynomials over infinite fields

This question concerns teaching a proof of the theorem that if a polynomial $f \in k[x]$ over an infinite field $k$ is the zero function (i.e. $f(a) = 0$ for all $a \in k$) then it is also the zero ...
J W's user avatar
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5 votes
1 answer
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Choice of textbook for an undergraduate abstract algebra course

Currently a 5th year PhD student, and I've been fighting tooth and nail to teach one of our junior-year honors sections in undergraduate algebra next fall (desperately hopeful we'll be able to return ...
jons_stupid_questions's user avatar
4 votes
6 answers
646 views

Applications of abstract algebra outside of mathematics and suitable textbook

The question What are some good mathematical applications to present in an abstract algebra course? asks about mathematical applications of abstract algebra. What are some applications of abstract ...
J W's user avatar
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3 votes
4 answers
245 views

How to naturally encounter the properties of identity, commutativity, associativity, and distributivity (to define rings)?

(Cross posted at MSE: https://math.stackexchange.com/questions/3742948/how-did-we-isolate-the-properties-of-identity-commutativity-associativity-and) In elementary school, I remember learning about ...
D.R's user avatar
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3 answers
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The term "unique" for functions and operations

This is long so... TLDR: Proposing the math community steer away from the misleading term unique, with respect to functions and algebraic operations. Instead, use unambiguous. Why not? Analysis below....
Zduff's user avatar
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13 votes
2 answers
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Introductory real analysis before or after introductory abstract algebra?

What are the pros and cons for students of taking introductory real analysis before or after introductory abstract algebra, assuming they are going to take both? I recognize that the overlap between ...
J W's user avatar
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8 votes
3 answers
2k views

MacLane-Birkhoff's "Algebra" vs Jacobson's "Basic Algebra I,II" vs Lang's "Algebra"

(Cross-posted at Math.Stackexchange) I'm searching for an apt textbook(s) on Abstract Algebra for a very advanced undergraduate/graduate level course in Algebra, and would be grateful for any help. ...
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2 votes
0 answers
280 views

Online open-course-ware that uses Maclane's book "Algebra"

I am struggling with that book which I find to be more of second-guessing type than a book for self-study: it has cryptically written sections, no examples (and those given, and rarely, are even more ...
André Gargoura's user avatar
5 votes
5 answers
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Where do students learn to solve polynomial equations these days?

When I was a math undergraduate 30 years ago in India, we were taught what was then called "classical algebra" (as opposed to abstract algebra), and we were taught among other things solving ...
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6 votes
4 answers
295 views

Group theory by geometry

I'm introducing my kids to the concepts of group theory. To make abstract things tangible, I'm trying the geometry way, adopting Arnold's in "Abel's Theorem", so far I've explained, by using symmetry ...
athos's user avatar
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12 votes
2 answers
307 views

Alternating group without $S_n$

I'm going to start introducing my abstract algebra class to a variety of groups soon. Dihedral groups $D_n$ arise out of symmetries on polygons. And the Symmetric group $S_n$ makes sense as the group ...
Aeryk's user avatar
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4 votes
3 answers
402 views

Geometry textbook with an abstract algebra emphasis

I'm teaching a variety of undergraduate and graduate geometry classes (mostly for in-service teachers) which range from elementary axiomatic geometry to more advanced transformational geometry. I'm ...
Jeremy Brazas's user avatar
4 votes
0 answers
319 views

According to Nathan Jacobson, what is Intermediate Algebra and Advanced Algebra?

Nathan Jacobson's Basic Algebra I, II covers many topics in Algebra that is probably even beyond many pure mathematics full professor's scope of knowledge, unless the professor is specialised in ...
Zuriel's user avatar
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6 votes
2 answers
292 views

At what point in the curriculum should the tensor product be introduced?

I remember my linear algebra teacher mentioning tensor products as an advanced topic that would be covered in upper level algebra coursework. During undergraduate abstract algebra, tensor products ...
Alexander Gruber's user avatar
4 votes
2 answers
208 views

A good example to show group actions and Burnside's lemma

I want to make a presentation of Burnside's lemma outside of group theory, and more as the stand-alone combinatorial tool that it can also be. My plan right now is to make it into a 15-20 minute video,...
Arthur's user avatar
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5 votes
1 answer
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Open Rings-First Abstract Algebra Text

Building off my own experience and the responses to "Rings before groups in abstract algebra?" I've decided to teach Abstract Algebra using a rings-first approach. However the various texts mentioned ...
Aeryk's user avatar
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3 votes
0 answers
75 views

Formal linear combinations: motivating examples

I want to introduce formal linear combinations in an upper-level undergraduate combinatorics class. By this I mean expressions like $7 \operatorname{cat} + 5 \operatorname{dog} - \sqrt{2} \...
darij grinberg's user avatar
3 votes
2 answers
307 views

Succinct description of situations where naively obvious is correct, but for far more complicated reasons?

What is the name for a situation where the obvious thing turns out to be true, but the reasoning is more complicated? In abstract algebra we can say the rational numbers - the fractions, $\mathbb{Q}...
john mangual's user avatar
6 votes
3 answers
273 views

What made (abstract) algebra grow in relative importance?

Nowadays, when I look at mathematics programs of study, "algebra" (at the abstract level) and "analysis" are treated as equally important. I'm "dating" myself, but this did not appear to be true in ...
Tom Au's user avatar
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13 votes
3 answers
2k views

How to teach abstract algebra for the first time?

I am a Ph.D student in computer science. I am TAing one course this semester, which requires the basics of abstract algebra like rings, fields, ideals, and basic theorems about them. I have done two ...
ddd's user avatar
  • 273
17 votes
8 answers
1k views

Examples of basic non-commutative rings

I am teaching an intro to ring theory, and after grading the first quiz, I realize most of my students are under the assumption that rings must be commutative. I have given them the example of ...
David Steinberg's user avatar
7 votes
1 answer
140 views

Looking for online abstract algebra courses making use of computer algebra systems

preferably areas of algebra of value and interest to computing practitioners. any level from introductory to (say) Grobner bases. preferably using open source computer algebra software. no preference ...
Bill Bell's user avatar
  • 173
7 votes
1 answer
219 views

Explaining genus to students

I need to do a presentation on my thesis, which is in arithmetic geometry. This presentation is meant for all students of mathematics, but I will assume some knowledge of abstract algebra (i.e. groups,...
Krijn's user avatar
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18 votes
1 answer
7k views

Is MacLane and Birkoff's "Algebra" suitable today as either an undergraduate or graduate text in abstract algebra?

I'm going to soon review the 3rd edition of Saunders MacLane And Garrett Birkoff's Algebra at my blog soon and this is the first time I'm really carefully reading it. While I'm really enjoying the ...
The Mathemagician's user avatar
6 votes
4 answers
379 views

Content for a two-quarter class on abstract algebra

I am going to teach a two-quarter sequence on abstract algebra at a mid-size american public university. Ideally, this course would introduce groups, rings, and fields, then end with some ...
David Steinberg's user avatar
11 votes
9 answers
637 views

Simple examples that violate group axioms

In a course for non-math-majors at a liberal arts college, I would like to give a few lectures and activities about groups and symmetry. I think it's straightforward to explain the group axioms and ...
Brendan W. Sullivan's user avatar
14 votes
3 answers
426 views

What are the "best" groups to use as examples while learning new concepts in algebra?

This question was asked to Math SE at first but it seems like it is more appropriate to ask it here. While learning new concepts in algebra it is quite helpful to check some examples which includes ...
Ninja's user avatar
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0 votes
1 answer
245 views

Comments on my approach to Group Theory notes?

I am working on some introductory notes for group theory. Comments on my initial approach here and any errors so far would be appreciated. I begin with the group axioms: $\forall a,b\in G:[ a+b\in ...
Dan Christensen's user avatar
41 votes
4 answers
5k views

Rings before groups in abstract algebra?

The default approach to teaching abstract algebra seems to be groups first, then rings. However, occasionally a textbook pops up (e.g. Childs' A Concrete Introduction to Higher Algebra, Hodge et al's ...
J W's user avatar
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8 votes
2 answers
221 views

When to cover other algebraic structures in an abstract algebra course?

I am preparing to give an abstract algebra course, which should be mainly focused on Group Theory. However, I want to cover other algebraic structures (magmas, quasigroups, etc.) as well. I am ...
wythagoras's user avatar
21 votes
10 answers
1k views

What makes cosets hard to understand?

I have been teaching introductory group theory to undergraduates. We reached cosets several weeks ago, but the combination of the textbook, my explanations and various practice questions has left the ...
Jessica B's user avatar
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21 votes
2 answers
1k views

Impossibility of trisecting the angle, doubling the cube and alike, what are reasons for or against discussing them in a course on algebra?

When I taught courses on algebra giving a first exposition to Galois theory I usually included some discussion of classical results showing the impossibility of constructing certain points with ruler ...
quid's user avatar
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3 votes
2 answers
560 views

Seeking your advice on books for abstract algebra and linear algebra

I am a college sophomore in the US with a major in mathematics and am an aspiring mathematician in the fields of computational complexity theory and cryptography. I would like to seek your advice and ...
MathWanderer's user avatar
12 votes
6 answers
881 views

Graphing functions from a finite field to itself

I have been teaching a ring theory course this semester, focusing on modular arithmetic and quotient rings of polynomials over fields. Several students have asked me how one could graph functions ...
Brian Rushton's user avatar
5 votes
5 answers
382 views

Notation for an element in a polynomial ring

Let $F$ be a field. What is the best notation (in an undergraduate or graduate abstract algebra class) for a generic element of the univariate polynomial ring $F[x]$? The most common notation seems ...
Frank Thorne's user avatar
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