# Tagged Questions

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### Working towards Abel's proof of unsolvability of quintics

I am currently doing a course in Abstract Algebra. I have been told that while some of the basic theory is laid down, we will not get as far as actually proving the unsolvability of quintics. ...
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### What area of Abstract Algebra do you find most interesting? [closed]

For my Abstract Algebra class, we will be doing small presentations (2 class periods) covering some topic in Abstract Algebra. Thus far, I have studied groups, rings, fields, modules, tensor ...
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### The motivation behind axiomatisation

Axiomatisation in the context of rings I am in the middle of an elementary pure mathematics unit and have just started looking at the concept of rings. In lectures, we have divided up rings into ...
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Reading math has gradually expanded my understanding of the word "symmetry", so that now I can recognize symmetries that I would have not noticed before, and without having them pointed out to me. I ...
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### Number Theory and Cryptography

I am a math tutor at a community college, and I stopped in to ask one of the professors a question about crypto and he lent me a graduate level book on for a full year course in the title of this ...
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### Intuition about structures in Galois Theory.

Background Often in mathematics I find we can ask "why" something is true. Of course, this is not a well defined question. However, it usually prompts answers that includes words like ...
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### Is it possible to learn abstract algebra no precalculus or calculus?

See above. I am trying to re-teach myself mathematics in a different manner than is formally taught (i.e., set theory, number theory, mathematical logic, abstract algebra, discrete math and then ...
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### How are theorems developed/evolved? How would have a mathematician like Cauchy thought about the following theorem?

I am not sure if i can ask this question over her but the other day i was studying this theorem If $\epsilon = \beta_1 \beta_2 ...... \beta_r$ where the $\beta's$ are 2- cycles, then r is even. The ...
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### Why teach linear algebra before abstract algebra?

Is there a reason why most undergraduate curriculums put linear algebra before abstract algebra? I'm asking this because personally it seems to be much easier to understand the architecture behind ...
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### Categorical introduction to Algebra and Topology

I am self-studying Mathematics in my free time. At the moment I am reading books on Algebra and on Category theory. More exactly, I started working through the book $\textit{Algebra}$ by Serge Lang. I ...
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### (soft) How hard REALLY is Advanced Calculus I and Abstract Algebra I [closed]

WTF everyone here downvote Arthur Fischer he is asking for it!!! This next fall, I will be taking Advanced Calculus I and Abstract Algebra I, along with Physical Chemistry I, Computer Science I, ...
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### Modular Arithmetic - Are we allowed to distribute the Modularity?

Assume I have a problem such as "Prove that $\displaystyle103^{53} + 53^{103}$ is divisible by $39$." This would mean I wanted to prove that $\displaystyle103^{53} + 53^{103}\equiv0\pmod{39}$. My ...
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### What is the “circle-plus” symbol I see in Abstract Algebra?

Sorry if this comes off as a random or soft question. I keep seeing this symbol in my abstract algebra course where it is a plus sign inside of a circle. I am not sure what it means. Can someone ...
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### What does quotienting by a congruence mean?

I have come across quotient algebras in my different mathematics courses. I know of quotienting with normal groups, quotienting with ideals etc. While studying Boolean Algebra I encounter quotienting ...
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### What fields (and operators acting on those fields) might form the basis of alien mathematics?

Addition and multiplication, according to most histories, arose in human civilizations out of a need to count a finite number of objects, and then later on especially, to measure land. What might be ...
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### Why are the surreals considered “recreational” mathematics?

One of my college math professors once remarked to me that it was interesting that John Conway's two "biggest" contributions to math were both recreational: the Game of Life and the Surreals. No one, ...
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### On which structures does the free group 'naturally' act?

One of the best ways to get a handle on a group is to recognize it as isomorphic to a set of symmetries of some structure. The dihedral group of order $2n$ is easily recognized as the set of ...
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### Which are the most effective modern intuitive definitions of a vector?

First, I would like to clarify what I mean by "intuitive definition": an intuitive definition is an informal understanding of a concept which helps to build mental agility, with the possible ...
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### What are some properties that imply that a group must be the trivial group?

In the problem posed in this question of mine we want to show that a particular group is both perfect and solvable, and therefore trivial, and this turns out to be useful in proving the result. What ...
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### Resources for self-learning “relational” abstract algebra? [please see body of post for details]

I have been studying Grassman and Clifford algebras a bit, and it is fascinating to see how, for example, the rules defining the inner product operator are enough to the capture something of the ...
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### Why are particular combinations of algebraic properties “better” (richer and more pervasive) than others?

I am a second-year graduate student of pure mathematics. Like most graduate students, I have been exposed to many types of algebraic structures, and it seems standard for the major emphasis, if not ...
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### Solving Abstract Problems

I'm doing this Solving Abstract Problem but I'm not sure which one it is. I mean from the Series I can see there's a pattern but in the Options I don't see images that link with the Series. Do you ...
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### Lang as a first algebra book

I think I am ready to learn algebra from Lang, but wanted some perspective. I have been exposed to: Linear algebra: All of Axler From my other, legendary honors course: -Order theory (lattices, ...
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### Obtaining a binary operation on $X \rightarrow Y$ from a binary operation on $Y$. What, if anything, to make of this observation?

Let $X$ and $Y$ denote sets. Then if $+$ is a binary operation on $Y$, then we can obtain a new binary operation $+'$ on $Y^X$ in a canonical way as follows. $$(f+' g)(x) = f(x)+g(x)$$ Question. The ...
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### “Why” is $[\mathbb{C}:\mathbb{R}] < \infty$?

Obviously this question is a little open-ended. A lot of complex analysis seems to work primarily because we can view $\mathbb{C}$ as a finite-dimensional $\mathbb{R}$-algebra, and apply analytic and ...
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### Tips for writing proofs

When writing formal proofs in abstract and linear algebra, what kind of jargon is useful for conveying solutions effectively to others? In general, how should one go about structuring a formal proof ...
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### Generic elementary group theory problems.

This question is about generic group theory problems. here are examples for what I’m referring to: Prove that any group of order $p^2$, where $p$ is a prime, is abelian. Let $G$ be a ...
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### Real life examples of commutative but non-associative operations

I've been trying to find ways to explain to people why associativity is important. Subtraction is a good example of something that isn't associative, but it is not commutative. So the best I could ...
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### Why do we have $3$ Isomorphism Theorems?

This is a bit of a soft question, but I've wondered about it since being introduced to the $3$ isomorphism theorems (I'm aware of the $4^{th}$ as well, but it is not typically presented in the ...
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### On the powerset of a monoid, does the second operation play nice with the first? Indeed, is it even useful?

Let $X$ denote a monoid. Then we can make $Y = \mathcal{P}(X)$ into a monoid, too. Define $$AB = \{ab \mid a \in A, b \in B\}$$ for all $A,B \in Y.$ We see immediately that $1$ (shorthand for $\{1\}$) ...
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### definitions in algebra

I am interested in algebra and I want to read some good books about it. I have a problem with some definitions like free group or algebra. In some books there are different definitions of them and I'm ...
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### Commutative/noncommutative algebra?

I know basic knowledge of undergraduate algebra till galois theory of finite extensions. I want to learn number theory, but also like algebra. This semester I have to choose to read either commutative ...
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### What's the exact meaning of this sentence from George Peacock?

I am reading the book "A history of abstract algebra" by Israel Kleiner. The following sentence is said by George Peacock. I am not a native English speaker. So could someone translate it into plain ...
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I know that : if two subgroups of $G$ are conjugate then they are isomorphic. Howerver , I also know that the converse is not always true. I often understand a mathematical structure only trough ...
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### Show a subring contains certain elements.

Show that the set of all real numbers of the form $a_0 + a_1\pi + a_2\pi^2 +\cdots+ a_n\pi^n$ with $n≥0$ and $a_i ∈ \mathbb{Z}$ is a subring of $R$ that contains $\mathbb{Z}$ and $\pi$. ...
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### Learning Abstract Algebra for a graduate degree

I would like to do a graduate degree in mathematics, and I have a full year before I will be able to do so (for personal reasons). I mainly have my weekends available to study. I am interested in ...
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### Abstract Algebra book with exercise solutions recommendations.

I am new to studying abstract algebra (and math in general). I've been reading Gilligan and Pinter's books .I am trying to improve my understanding by doing exercises. However none of the books I am ...
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### Could we reasonably classify finite groups?

So I have been reading some work on $p$-groups, and I noticed a particularly disturbing sentence: "There is no hope of finding a finite set of invariants that will define every p-group up to ...
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### Is there a geometric idea behind Sylow's theorems?

I have a confession to make: none of the proofs of Sylow's theorems I saw clicked with me. My first abstract algebra courses were more on the algebraic side (without mention of group actions and ...
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### how can we write abstract algorithms?

Writing pseudo-code for algorithms is common practice in the applied mathematics literature. It is also often the case that the ideal input of an algorithm is an infinite set, for example it could be ...
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### interesting topic about representation theory

I have to develop a discussion about representation theory (about 30 pages). My knowledge is very superficial and limited to general representations theory of groups and characters theory. Do you know ...
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### 4 big ideas in algebra that have rich connections to other fields? [closed]

In his controversial post criticizing high school algebra, Grant Wiggins issued a challenge to his readers: Can you identify 4 big ideas in algebra, ideas that not only provide a powerful set of ...
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### Theorems with the greatest impact on group theory as a whole

In his Contemporary Abstract Algebra text, Gallian asserts that Sylow's Theorem(s) and Lagrange's Theorem are the two most important results in finite group theory. He also provides this quote by ...
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### What if $\pi$ was an algebraic number? (significance of algebraic numbers)

To be honest, I never really understood the importance of algebraic numbers. If we lived in an universe where $\pi$ was algebraic, would there be a palpable difference between that universe and ours? ...
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### categorical generalizations of familiar objects

A couple of days ago I've learned that you can define trace in a very abstract setting. Namely, suppose $F\colon A\to B$ is a functor between two categories. Suppose $E,G\colon B\to A$ are two ...
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### What's the motivation to add inner product and wedge product together in geometric product

I am reading some geometric algebra notes. They all started from some axioms. But I am still confused on the motivation to add inner product and wedge product together by defining  ab = a\cdot b + ...
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### What sections should I study to prove that fifth (and up) degree polynomial equations are not solvable with Fraleigh?

I'm Korean high school student who wants to study how to prove that degree ≥5 polynomial equations are not solvable. I know some of Set Theory and will study abstract algebra with 'A First Course in ...
Given an integral domain $D$, there are several ways how to construct a new integral domain related to D. For example, one can consider a ring of polynomials/formal power series/formal Laurent series ...