0
votes
1answer
49 views

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. ...
2
votes
5answers
298 views

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 ...
2
votes
1answer
43 views

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 ...
9
votes
3answers
113 views

How to recognize adjointness?

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 ...
1
vote
2answers
80 views

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 ...
5
votes
0answers
86 views

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 ...
1
vote
3answers
138 views

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 ...
0
votes
0answers
41 views

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 ...
11
votes
5answers
2k views

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 ...
6
votes
3answers
171 views

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 ...
-5
votes
2answers
142 views

(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, ...
3
votes
5answers
293 views

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 ...
0
votes
1answer
67 views

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 ...
2
votes
1answer
61 views

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 ...
5
votes
2answers
439 views

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 ...
5
votes
3answers
212 views

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, ...
11
votes
1answer
258 views

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 ...
3
votes
6answers
276 views

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 ...
6
votes
3answers
205 views

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 ...
1
vote
0answers
36 views

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 ...
10
votes
4answers
146 views

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 ...
1
vote
2answers
42 views

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 ...
3
votes
1answer
161 views

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, ...
6
votes
1answer
76 views

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 ...
10
votes
1answer
209 views

“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 ...
12
votes
4answers
269 views

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 ...
20
votes
6answers
324 views

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 ...
6
votes
4answers
451 views

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 ...
3
votes
3answers
77 views

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 ...
3
votes
2answers
70 views

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 $\{1\}$) ...
2
votes
3answers
81 views

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 ...
5
votes
2answers
114 views

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 ...
2
votes
2answers
163 views

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 ...
1
vote
1answer
76 views

question about concepts: isomorphic,conjugate.

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 ...
0
votes
1answer
85 views

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 $\pi$. ...
9
votes
2answers
599 views

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 ...
3
votes
3answers
2k views

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 ...
9
votes
1answer
155 views

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 ...
18
votes
1answer
370 views

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 ...
1
vote
3answers
100 views

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 ...
2
votes
2answers
113 views

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 ...
2
votes
1answer
187 views

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 ...
12
votes
2answers
492 views

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 ...
38
votes
11answers
2k views

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? ...
2
votes
1answer
124 views

How is it shown that quintic equations can be solved by radicals and ultraradicals?

See this article from wikipedia: http://en.wikipedia.org/wiki/Bring_radical George Jerrard showed that some quintic equations can be solved using radicals and Bring radicals, which had been ...
10
votes
3answers
290 views

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 ...
2
votes
3answers
265 views

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 + ...
1
vote
1answer
90 views

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 ...
2
votes
0answers
48 views

Is this slightly different proof to Hibert's Theorem “different enough”?

I generally try and think up slightly different proofs to the material that I read in order to grasp some deeper possible insight, and then painstakingly record them in a latex document. I've been ...
8
votes
1answer
179 views

Construction(s) of new integral domains from “old ones”

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 ...