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Questions tagged [abstract-algebra]

For questions about monoids, groups, rings, modules, fields, vector spaces, algebras over fields, various types of lattices, and other such algebraic objects. Associate with related tags like [group-theory], [ring-theory], [modules], etc. as necessary to clarify which topic of abstract algebra is most related to your question and help other users when searching.

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519 votes
7 answers

"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 $f_n(x)$ ...
Alexander Gruber's user avatar
  • 27.2k
332 votes
31 answers

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 lecture's script there are only examples like $\mathbb{Z}$ under addition and other things like that. I don't ...
256 votes
1 answer

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.
leo's user avatar
  • 10.5k
227 votes
11 answers

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 ...
201 votes
7 answers

Do we have negative prime numbers?

Do we have negative prime numbers? $..., -7, -5, -3, -2, ...$
user avatar
185 votes
2 answers

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 $G\rightarrow H$?
Kerry's user avatar
  • 1,859
180 votes
6 answers

An Introduction to Tensors

As a physics student, I've come across mathematical objects called tensors in several different contexts. Perhaps confusingly, I've also been given both the mathematician's and physicist's definition, ...
Noldorin's user avatar
  • 6,668
177 votes
1 answer

Does every ring of integers sit inside a ring of integers that has a power basis?

Given a finite extension of the rationals, $K$, we know that $K=\mathbb{Q}[\alpha]$ by the primitive element theorem, so every $x \in K$ has the form $$x = a_0 + a_1 \alpha + \cdots + a_n \alpha^n,$$ ...
Eins Null's user avatar
  • 2,177
175 votes
6 answers

What are the differences between rings, groups, and fields?

Rings, groups, and fields all feel similar. What are the differences between them, both in definition and in how they are used?
cobbal's user avatar
  • 2,045
166 votes
3 answers

The square roots of different 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 ...
user8465's user avatar
  • 1,763
165 votes
2 answers

Example of infinite field of characteristic $p\neq 0$

Can you give me an example of infinite field of characteristic $p\neq0$? Thanks.
Aspirin's user avatar
  • 5,699
158 votes
1 answer

Classification of prime ideals of $\mathbb{Z}[X]$

Let $\mathbb{Z}[X]$ be the ring of polynomials in one variable over $\Bbb Z$. My question: Is every prime ideal of $\mathbb{Z}[X]$ one of following types? If yes, how would you prove this? $(0)$. $(...
Makoto Kato's user avatar
  • 42.8k
152 votes
41 answers

Why is negative times negative = positive?

Someone recently asked me why a negative $\times$ a negative is positive, and why a negative $\times$ a positive is negative, etc. I went ahead and gave them a proof by contradiction like this: ...
Sev's user avatar
  • 2,123
151 votes
4 answers

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 $...
Richard G 's user avatar
  • 3,925
145 votes
3 answers

How to find the Galois group of a polynomial?

I've been learning about Galois theory recently on my own, and I've been trying to solve tests from my university. Even though I understand all the theorems, I seem to be having some trouble with the ...
IBS's user avatar
  • 4,215
138 votes
16 answers

Why is $1$ not a prime number?

Why is $1$ not considered a prime number? Or, why is the definition of prime numbers given for integers greater than $1$?
bryn's user avatar
  • 9,784
135 votes
6 answers

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 $\mathbb C$,...
MJD's user avatar
  • 65.7k
134 votes
0 answers

If polynomials are almost surjective over a field, is the field algebraically closed?

Let $K$ be a field. Say that polynomials are almost surjective over $K$ if for any nonconstant polynomial $f(x)\in K[x]$, the image of the map $f:K\to K$ contains all but finitely many points of $K$. ...
Eric Wofsey's user avatar
133 votes
8 answers

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 ...
Srivatsan's user avatar
  • 26.4k
132 votes
9 answers

Normal subgroup of prime index

Generalizing the case $p=2$ we would like to know if the statement below is true. Let $p$ the smallest prime dividing the order of $G$. If $H$ is a subgroup of $G$ with index $p$ then $H$ is normal.
Sigur's user avatar
  • 6,466
128 votes
8 answers

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 ...
Vicfred's user avatar
  • 2,797
126 votes
7 answers

Why are There No "Triernions" (3-dimensional analogue of complex numbers / quaternions)? [duplicate]

Since there are complex numbers (2 dimensions) and quaternions (4 dimensions), it follows intuitively that there ought to be something in between for 3 dimensions ("triernions"). Yet no one uses ...
thecat's user avatar
  • 1,848
124 votes
7 answers

Are all algebraic integers with absolute value 1 roots of unity?

If we have an algebraic number $\alpha$ with (complex) absolute value $1$, it does not follow that $\alpha$ is a root of unity (i.e., that $\alpha^n = 1$ for some $n$). For example, $(3/5 + 4/5 i)$ ...
Jonas Kibelbek's user avatar
124 votes
3 answers

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? ...
Hui Yu's user avatar
  • 15.1k
123 votes
13 answers

Good abstract algebra books for self study

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,...
121 votes
12 answers

Is there an "inverted" dot product?

The dot product of vectors $\mathbf{a}$ and $\mathbf{b}$ is defined as: $$\mathbf{a} \cdot \mathbf{b} =\sum_{i=1}^{n}a_{i}b_{i}=a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}$$ What about the quantity? $$\...
doc's user avatar
  • 1,307
116 votes
8 answers

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 ...
Alexander Gruber's user avatar
  • 27.2k
114 votes
13 answers

Does commutativity imply Associativity?

Does commutativity imply associativity? I'm asking this because I was trying to think of structures that are commutative but non-associative but couldn't come up with any. Are there any such examples? ...
Eugene's user avatar
  • 7,652
114 votes
0 answers

Ring structure on the Galois group of a finite field

Let $F$ be a finite field. There is an isomorphism of topological groups $(\mathrm{Gal}(\overline{F}/F),\circ) \cong (\widehat{\mathbb{Z}},+)$. It follows that the Galois group carries the structure ...
Martin Brandenburg's user avatar
113 votes
6 answers

Finite subgroups of the multiplicative group of a field are cyclic

In Grove's book Algebra, Proposition 3.7 at page 94 is the following If $G$ is a finite subgroup of the multiplicative group $F^*$ of a field $F$, then $G$ is cyclic. He starts the proof by ...
QETU's user avatar
  • 1,139
107 votes
13 answers

Why would I want to multiply two polynomials?

I'm hoping that this isn't such a basic question that it gets completely laughed off the site, but why would I want to multiply two polynomials together? I flipped through some algebra books and ...
user3818's user avatar
  • 1,079
107 votes
2 answers

A semigroup $X$ is a group iff for every $g\in X$, $\exists! x\in X$ such that $gxg = g$

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 ...
Lubin's user avatar
  • 63.1k
105 votes
1 answer

In categorical terms, why is there no canonical isomorphism from a finite dimensional vector space to its dual?

I've read in several places that one motivation for category theory was to be able to give precise meaning to statements like, "finite dimensional vector spaces are canonically isomorphic to their ...
Ben Blum-Smith's user avatar
103 votes
8 answers

How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$?

Let $p$ be a prime. How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$? Right now I'm able to prove that it has no roots and that it is separable, but I have not ...
MathTeacher's user avatar
  • 1,549
103 votes
7 answers

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 ...
user avatar
102 votes
4 answers

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 ...
Bruno Stonek's user avatar
  • 12.6k
100 votes
8 answers

Intuitive meaning of Exact Sequence

I'm currently learning about exact sequences in grad sch Algebra I course, but I really can't get the intuitive picture of the concept and why it is important at all. Can anyone explain them for me? ...
finnlim's user avatar
  • 2,743
95 votes
6 answers

Why can't the Polynomial Ring be a Field?

I'm currently studying Polynomial Rings, but I can't figure out why they are Rings, not Fields. In the definition of a Field, a Set builds a Commutative Group with Addition and Multiplication. This ...
IAE's user avatar
  • 1,347
95 votes
3 answers

Prove that the set of all algebraic numbers is countable

A complex number $z$ is said to be algebraic if there are integers $a_0, ..., a_n$, not all zero, such that $a_0z^n+a_1z^{n-1}+...+a_{n-1}z+a_n=0$. Prove that the set of all algebraic numbers is ...
PandaMan's user avatar
  • 3,229
94 votes
5 answers

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 $\...
Leo's user avatar
  • 10.7k
94 votes
8 answers

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 ...
user avatar
94 votes
7 answers

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) ...
Michael Benfield's user avatar
93 votes
6 answers

Is $\mathbb{Q}(\sqrt{2}, \sqrt{3}) = \mathbb{Q}(\sqrt{2}+\sqrt{3})$?

Is $\mathbb{Q}(\sqrt{2}, \sqrt{3}) = \mathbb{Q}(\sqrt{2}+\sqrt{3})$ ? $$\mathbb{Q}(\sqrt{2},\sqrt{3})=\{a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6} \mid a,b,c,d\in\mathbb{Q}\}$$ $$\mathbb{Q}(\sqrt{2}+\sqrt{3}) = ...
Tashi's user avatar
  • 1,633
91 votes
4 answers

If $G/Z(G)$ is cyclic, then $G$ is abelian

Continuing my work through Dummit & Foote's "Abstract Algebra", 3.1.36 asks the following (which is exactly the same as exercise 5 in this related MSE answer): Prove that if $G/Z(G)$ is cyclic, ...
Altar Ego's user avatar
  • 5,342
90 votes
2 answers

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 ...
lhf's user avatar
  • 217k
88 votes
3 answers

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) = (a*b)...
Mikko Korhonen's user avatar
88 votes
9 answers

Terence Tao–type books in other fields?

I have looked at Tao's book on Measure Theory, and they are perhaps the best math books I have ever seen. Besides the extremely clear and motivated presentation, the main feature of the book is that ...
85 votes
7 answers

Quotient ring of Gaussian integers

A very basic ring theory question, which I am not able to solve. How does one show that $\mathbb{Z}[i]/(3-i) \cong \mathbb{Z}/10\mathbb{Z}$. Extending the result: $\mathbb{Z}[i]/(a-ib) \cong \mathbb{...
user avatar
84 votes
5 answers

How to show that the commutator subgroup is a normal subgroup

It is suggested as an exercise in Serge Lang's book "Algebra" to show that the commutator subgroup $G^c$ of a group $G$ is a normal subgroup. I'd like to do that but I am afraid I need help, I ...
harlekin's user avatar
  • 8,790
83 votes
3 answers

Polynomials irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$

Let $f(x) \in \mathbb{Z}[x]$. If we reduce the coefficents of $f(x)$ modulo $p$, where $p$ is prime, we get a polynomial $f^*(x) \in \mathbb{F}_p[x]$. Then if $f^*(x)$ is irreducible and has the same ...
spin's user avatar
  • 12k

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