# Questions tagged [abstract-algebra]

For questions about groups, rings, fields, vector spaces, modules and other algebraic objects. Associate with related tags like group-theory, ring-theory, modules, etc. to clarify which topic of abstract algebra is most related to your question and help other users when searching.

57,554 questions
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### “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 $f_n(x)$ ...
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### 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 don't ...
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### 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.
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### 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 ...
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### 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 $G\rightarrow H$?
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### 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, ...
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### 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 ...
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 $... 12answers 11k views ### Is there an “inverted” dot product? The dot product of vectors$\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? $$\... 2answers 21k views ### Example of infinite field of characteristic p\neq 0 Can you give me an example of infinite field of characteristic p\neq0? Thanks. 8answers 17k views ### When to learn category theory? I'm a undergraduate who wishes to learn category theory but I only have basic knowledge of linear algebra and set theory, I've also had a short course on number theory which used some basic concepts ... 5answers 6k views ### Why are the solutions of polynomial equations so unconstrained over the quaternions? An nth-degree polynomial has at most n distinct zeroes in the complex numbers. But it may have an uncountable set of zeroes in the quaternions. For example, x^2+1 has two zeroes in \mathbb C,... 7answers 7k views ### Why are There No “Triernions” (3-dimensional analogue of complex numbers / quaternions)? [duplicate] Since there are complex numbers (2 dimensions) and quaternions (4 dimensions), it follows intuitively that there ought to be something in between for 3 dimensions ("triernions"). Yet no one uses ... 6answers 6k views ### Why “characteristic zero” and not “infinite characteristic”? The characteristic of a ring (with unity, say) is the smallest positive number n such that$$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$provided such an n exists. Otherwise, we ... 2answers 8k 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? ... 6answers 62k views ### 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? 8answers 25k 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 ... 3answers 3k views ### 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 ... 3answers 28k views ### 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 ... 6answers 3k 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 ... 7answers 8k 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 ... 5answers 11k views ### 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) ... 1answer 12k views ### 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? ... 7answers 7k 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 ... 4answers 4k 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 ... 7answers 6k 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) ... 7answers 8k views ### What is lost when we move from reals to complex numbers? [duplicate] As I know when you move to "bigger" number systems (such as from complex to quaternions) you lose some properties (e.g. moving from complex to quaternions requires loss of commutativity), but does it ... 5answers 7k 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 \... 1answer 1k views ### Ring structure on the Galois group of a finite field Let F be a finite field. There is an isomorphism of topological groups \left(\mathrm{Gal}(\overline{F}/F),\circ\right) \cong (\widehat{\mathbb{Z}},+). It follows that the Galois group carries the ... 7answers 5k views ### Does associativity imply commutativity? I used to think that commutativity and associativity are two distinct properties. But recently, I started thinking of something which has troubled this idea:$$(1+1)+1 = 1+ (1+1)\implies 2+1=1+2$$... 2answers 3k views ### Is there an intuitive reason for a certain operation to be associative? When I read Pinter's A Book of Abstract Algebra, Exercise 7 on page 25 asks whether the operation$$x*y=\frac{xy}{x+y+1}$$(defined on the positive real numbers) is associative. At first I considered ... 1answer 4k 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 ... 14answers 4k views ### What are some mathematical topics that involve adding and multiplying pictures? Let me give you an example of what I mean. Flag algebras are a tool used in extremal graph theory which involve writing inequalities that look like: (It's not too important to my question what this ... 7answers 15k views ### 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 ... 1answer 3k 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 ... 8answers 5k views ### 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? ... 12answers 9k views ### 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? ... 10answers 26k views ### Examples of finite nonabelian groups. Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups? 5answers 30k views ### Is \mathbf{Q}(\sqrt{2}, \sqrt{3}) = \mathbf{Q}(\sqrt{2}+\sqrt{3})? [duplicate] Is \mathbf{Q}(\sqrt{2}, \sqrt{3}) = \mathbf{Q}(\sqrt{2}+\sqrt{3}) ?$$\mathbf{Q}(\sqrt{2},\sqrt{3})=\{a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6} \mid a,b,c,d\in\mathbf{Q}\}\mathbf{Q}(\sqrt{2}+\sqrt{3})... 1answer 2k views ### 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 ... 3answers 6k 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,$*is associative and distributive: \begin{align*} &a * (b * c) = (a*b)... 3answers 6k views ### Polynomials irreducible over\mathbb{Q}$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 ... 4answers 4k 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 ... 3answers 2k 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 ... 1answer 4k views ### Abstract nonsense proof of snake lemma During my studies, I always wanted to see a "purely category-theoretical" proof of the Snake Lemma, i.e. a proof that constructs all morphisms (including the snake) and proves exactness via universal ... 6answers 23k views ### 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 ... 3answers 4k views ### If I know the order of every element in a group, do I know the group? Suppose$G$is a finite group and I know for every$k \leq |G|$that exactly$n_k$elements in$G$have order$k$. Do I know what the group is? Is there a counterexample where two groups$G$and$H$... 10answers 74k views ### 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,... 6answers 13k views ### 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{...
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.