Abstract algebra is a study of algebraic objects. Some of the more common algebraic objects are Groups, Rings, Fields, Vector spaces, Modules, and other advanced topics.
156
votes
6answers
4k views
“The Egg:” Bizarre behavior of the roots of a family of polynomials.
In this MO post, I ran into the following family of polynomials: $$f_n(x)=\sum_{m=0}^{n}\prod_{k=0}^{m-1}\frac{x^n-x^k}{x^m-x^k}.$$
In the context of the post, $x$ was a prime number, and $f_n(x)$ ...
103
votes
18answers
8k views
Nice examples of groups which are not obviously groups
I am searching for some groups, where it is not so obvious that they are groups.
In the lectures script there are only examples like $\mathbb{Z}$ under
addition and other things like that. I ...
80
votes
2answers
2k views
Can we ascertain that there exists an epimorphism $G\rightarrow H$?
Let $G,H$ be finite groups. Suppose we have an epimorphism $$G\times G\rightarrow H\times H$$ Can we find an epimorphism $G\rightarrow H$?
74
votes
6answers
3k views
Does a “cubic” matrix exist?
Well, I've heard that a "cubic" matrix would exist and I thought: would it be like a magic cube? And more: does it even have a determinant - and other properties? I'm a young student, so... please ...
52
votes
4answers
2k views
Does $R[x] \cong S[x]$ imply $R \cong S$?
This is a very simple question but I believe it's nontrivial.
I would like to know if the following is true: If $R$ and $S$ are rings and $R[x]$ and $S[x]$ are isomorphic as rings, then $R$ and $S$ ...
52
votes
9answers
2k views
How do I sell out with abstract algebra?
My plan as an undergraduate was unequivocally to be a pure mathematician, working as an algebraist as a bigshot professor at a bigshot university. I'm graduating this month, and I didn't get into ...
49
votes
1answer
4k views
Why are rings called rings?
I've done some search in Internet and other sources about this question. Why the name ring to this particular object? Just curiosity.
Thanks.
48
votes
7answers
3k views
Why do books titled “Abstract Algebra” mostly deal with groups/rings/fields?
As a computer science graduate who had only a basic course in abstract algebra, I want to study some abstract algebra in my free time. I've been looking through some books on the topic, and most seem ...
48
votes
2answers
4k views
More than 99% of groups of order less than 2000 are of order 1024?
In Algebra: Chapter 0, the author made a remark (footnote on page 82), saying that more than 99% of groups of order less than 2000 are of order 1024.
Is this for real? How can one deduce this result? ...
48
votes
4answers
1k views
How is a group made up of simple groups?
I've read more than once the analogy between simple groups and prime numbers, stating that any group is built up from simple groups, like any number is built from prime numbers.
I've recently started ...
45
votes
1answer
1k views
How was the Monster's existence originally suspected?
I've read in many places that the Monster group was suspected to exist before it was actually proven to exist, and further that many of its properties were deduced contingent upon existence.
For ...
42
votes
8answers
2k views
Intuition in algebra?
My algebra background: I've had 2 undergrad semesters of algebra, a reading course in Galois Theory, a graduate course in commutative algebra and one in algebraic geometry, and I've done (most of) ...
40
votes
4answers
2k views
Why “characteristic zero” and not “infinite characteristic”?
The characteristic of a ring (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. Otherwise, we ...
40
votes
5answers
1k views
Why are groups more important than semigroups?
This is an open-ended question, as is probably obvious from the title. I understand that it may not be appreciated and I will try not to ask too many such questions. But this one has been bothering me ...
39
votes
8answers
732 views
Are there real world applications of finite group theory?
I would like to know whether there are examples where finite group theory can be directly applied to solve real world problems outside of mathematics. (Sufficiently applied mathematics such as ...
38
votes
6answers
2k views
What kind of “symmetry” is the symmetric group about?
There are two concepts which are very similar literally in abstract algebra: symmetric group and symmetry group. By definition, the symmetric group on a set is the group consisting of all bijections ...
37
votes
5answers
784 views
Why are the solutions of polynomial equations so unconstrained over the quaternions?
An $n$th-degree polynomial has at most $n$ distinct zeroes in the complex numbers. But it may have an uncountable set of zeroes in the quaternions. For example, $x^2+1$ has two zeroes in $\mathbb ...
34
votes
4answers
846 views
Algebra: Best mental images
I'm curious how people think of Algebras (in the universal sense, i.e., monoids, groups, rings, etc.). Cayley diagrams of groups with few generators are useful for thinking about group actions on ...
34
votes
2answers
810 views
Algebraic Topology Challenge: Homology of an Infinite Wedge of Spheres
So the following comes to me from an old algebraic topology final that got the best of me. I wasn't able to prove it due to a lack of technical confidence, and my topology has only deteriorated since ...
33
votes
1answer
1k views
Is Lagrange's theorem the most basic result in finite group theory?
Motivated by this question, can one prove that the order of an element in a finite group divides the order of the group without using Lagrange's theorem? (Or, equivalently, that the order of the group ...
32
votes
1answer
624 views
Is $1+x+\frac{x^2}2+\dots+\frac{x^n}{n!}$ irreducible?
The polynomial $f(x)=1+x+\frac{x^2}2+\dots+\frac{x^n}{n!}$ often appears in algebra textbooks as an illustration for using derivative to test for multiple roots.
Recently, I stumbled upon Example ...
32
votes
4answers
1k views
What kind of work do modern day algebraists do?
Often times in my studies I get the impression that algebra is just a tool to help with other branches of mathematics, like algebraic geometry, algebraic number theory, algebraic topology, etc. How ...
30
votes
8answers
3k views
When to learn category theory?
I'm a undergraduate who wishes to learn category theory but I only have basic knowledge of linear algebra and set theory, I've also had a short course on number theory which used some basic concepts ...
30
votes
3answers
531 views
Why do we look at morphisms?
I am reading some lecture notes and in one paragraph there is the following motivation: "The best way to study spaces with a structure is usually to look at the maps between them preserving structure ...
30
votes
7answers
570 views
Appearance of Formal Derivative in Algebra
When studying polynomials, I know it is useful to introduce the concept of a formal derivative. For example, over a field, a polynomial has no repeated roots iff it and its formal derivative are ...
30
votes
1answer
1k views
Is War necessarily finite?
War is an cardgame played by children and drunk college students which involves no strategic choices on either side. The outcome is determined by the dealing of the cards. These are the rules.
A ...
28
votes
2answers
2k views
The square roots of the primes are linearly independent over the field of rationals
I need to find a way of proving that the square roots of a finite set
of different primes are linearly independent over the field of
rationals. I've tried to solve the problem using elementary ...
28
votes
3answers
962 views
linear algebra over a division ring vs. over a field
When I was studying linear algebra in the first year, from what I remember, vector spaces were always defined over a field, which was in every single concrete example equal to either $\mathbb{R}$ or ...
27
votes
0answers
1k views
Does a four-variable analog of the Hall-Witt identity exist?
Lately I have been thinking about commutator formulas, sparked by rereading the following paragraph in Isaacs (p.125):
An amazing commutator formula is the Hall-Witt identity: ...
25
votes
4answers
1k views
Can someone explain the Yoneda Lemma to an applied mathematician?
I have trouble following the category-theoretic statement and proof of the Yoneda Lemma. Indeed, I followed a category theory course for 4-5 lectures (several years ago now) and felt like I understood ...
25
votes
2answers
563 views
Reference request for tricky problem in elementary group theory
The following could have shown up as an exercise in a basic Abstract Algebra text, and if anyone can give me a reference, I will be most grateful.
Consider a set $X$ with an associative law of ...
25
votes
4answers
880 views
“Cayley's theorem” for Lie algebras?
Groups can be defined abstractly as sets with a binary operation satisfying certain identities, or concretely as a collection of permutations of a set. Cayley's theorem ensures that these two ...
25
votes
2answers
402 views
When is the product of $n$ subgroups a subgroup?
Let $G$ be any group. It's a well-known result that if $H, K$ are subgroups of $G$, then $HK$ is a subgroup itself if and only if $HK = KH$.
Now, I've always wondered about a generalization of this ...
25
votes
0answers
663 views
What is the solution to Nash's problem presented in “A Beautiful Mind”?
I was watching the said movie the other night, and I started thinking about the equation posed by Nash in the movie. More specifically, the one he said would take some students a lifetime to solve ...
24
votes
10answers
1k views
Contributions of Galois Theory to Mathematics
What are the major and minor contributions of Galois Theory to Mathematics? I mean direct contributions (like being aplied as it appears in Algebra) or simply by serving as a model to other theories.
...
24
votes
3answers
497 views
Where does the word “torsion” in algebra come from?
Torsion is used to refer to elements of finite order under some binary operation. It doesn't seem to bear any relation to the ordinary everyday use of the word or with its use in differential geometry ...
23
votes
7answers
1k views
How to tell if some power of my integer matrix is the identity?
Given an $n\times n$-matrix $A$ with integer entries, I would like to decide whether there is some $m\in\mathbb N$ such that $A^m$ is the identity matrix.
I can solve this by regarding $A$ as a ...
23
votes
3answers
622 views
Galois groups of polynomials and explicit equations for the roots
Lets say I have calculated the galois group of some polynomial and I also have the subgroup structure. What's an effective procedure to turn the group into equations for the actual roots of the ...
22
votes
4answers
1k views
Center-commutator duality
I'm reading this article by Keith Conrad, on subgroup series. I'm having trouble with a statement he does at page 6:
Any subgroup of $G$ which contains $[G,G]$ is normal in $G$.
He says this as ...
22
votes
1answer
497 views
Why is the Hessian of an irreducible polynomial not zero?
Let $k$ be an algebraically closed field, $\operatorname{char}k=0$, $F$ be an irreducible homogeneous polynomial of degree$>1$ in $k[X,Y,Z]$, and ...
21
votes
3answers
2k views
Does multiplying polynomials ever decrease the number of terms?
Let $p$ and $q$ be polynomials (maybe in several variables, over a field), and suppose they have $m$ and $n$ non-zero terms respectively. We can assume $m\leq n$. Can it ever happen that the product ...
21
votes
6answers
535 views
Is $\mathbf{C}$ the algebraic closure of any field other than $\mathbf{R}$?
It seems to me (intuitively) that there should be no other fields whose algebraic closure is $\mathbf{C}$, even though I have no reason to believe it. The facts I've been using to formulate an ...
21
votes
6answers
890 views
Should I be worried that I am doing well in Analysis and not well in Algebra?
I attend a mostly liberal arts focused university, in which I was able to test out of an "Introduction to Proofs" class and directly into "Advanced Calculus 1" (Introductory Analysis I) and I loved ...
21
votes
4answers
627 views
Is an integer uniquely determined by its multiplicative order mod every prime
Let $x$ and $y$ be nonzero integers and $\mathrm{ord}_p(w)$ be the multiplicative order of $w$ in $ \mathbb{Z} /p \mathbb{Z} $. If $\mathrm{ord}_p(x) = \mathrm{ord}_p(y)$ for all primes (Edit: not ...
21
votes
3answers
437 views
On the meaning of being algebraically closed
The definition of algebraic number is that $\alpha$ is an algebraic number if there is a nonzero polynomial $p(x)$ in $\mathbb{Q}[x]$ such that $p(\alpha)=0$.
By algebraic closure, every nonconstant ...
21
votes
2answers
1k views
Does the equation $x^4+y^4+1 = z^2$ have a non-trivial solution?
The background of this question is this: Fermat proved that the equation,
$$x^4+y^4 = z^2$$
has no solution in the positive integers. If we consider the near-miss,
$$x^4+y^4-1 = z^2$$
then this has ...
21
votes
2answers
220 views
Can commuting matrices $X,Y$ always be written as polynomials of some matrix $A$?
Consider square matrices over a field $K$. I don't think additional assumptions about $K$ like algebraically closed or characteristic $0$ are pertinent, but feel free to make them for comfort. For any ...
21
votes
1answer
298 views
Lower bounds on the number of elements in Sylow subgroups
Let $p$ be a prime and $n \geq 1$ some integer. Furthermore, let $G$ be a finite group where $p$-Sylow subgroups have order $p^n$. Denote by $n_p(G)$ the number of Sylow $p$-subgroups of $G$. Denote ...
20
votes
9answers
3k views
$\sqrt a$ is either an integer or an irrational number.
I got this interesting question in my mind:
How do we prove that if $a \in \mathbb N$, then $\sqrt a$ is an integer or an irrational number?
Can we extend this result? That is, can it be shown ...
20
votes
3answers
995 views
Does every Abelian group admit a ring structure?
Given some Abelian group $(G, +)$, does there always exist a binary operation $*$ such that $(G, +, *)$ is a ring? That is, $*$ is associative and distributive:
\begin{align*}
&a * (b * c) = ...
