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4
votes
1answer
330 views

Mechanically instantiating abstract constructions

I am looking for work on the effective inverse of abstraction, aka specialization. There are two ways in which abstraction helps us: Get a better understanding of the structural rules at play in ...
12
votes
5answers
1k views

Are rings really more fundamental objects than semi-rings?

The discovery (or invention) of negatives, which happened several centuries ago by the Chinese, Indians and Arabs, has of course be of fundamental importance to mathematics. From then on, it seems ...
1
vote
1answer
195 views

How to distinguish property of particular representation from property of algebraic structure?

It is common that You have some interesting object (set, group, algebra or something, whatever) which has certain properties, structure etc. You may try describe it in pure algebraic way. Sometimes ...
6
votes
1answer
2k views

On order of subgroups in abelian groups

I wonder whether any of you guys has already read the homonymous note by R. Beals in the December 2009 issue of the Monthly. If so, would you be so kind as to let me know about the main ideas in ...
6
votes
8answers
4k views

Best way to teach concept of real numbers using a hands-on activity?

I know a middle school math teacher looking for some suggestions for hands-on activities to teach the concept of real numbers. I'm new to this site, so this may be a little off topic.
3
votes
1answer
267 views

Why are relations of degree 3 or less enough in a presentation of the polynomial current Lie algebra g[t]?

Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbb{C}$. The polynomial current Lie algebra $\mathfrak{g}[t] = \mathfrak{g} \otimes \mathbb{C} [t]$ has the bracket $$[xt^r, ...
2
votes
2answers
638 views

about state-field correspondence

In the defination of vertex algebra,we call the vertex operator state-field correspondence,does that mean that it is an injective map?? Are there some physical intepretations about state-field ...
3
votes
2answers
1k views

Using Weierstrass’s Factorization Theorem

I am trying to factorize $\sin(x)\over x$ which by Taylor series expansion and using the roots is $$a \cdot \left(1 - \frac{x}{\pi} \right) \left(1 + \frac{x}{\pi} \right) \left(1 - \frac{x}{2\pi} ...
2
votes
1answer
156 views

Introduction to modular property of affine alegebra and conformal vertex algebra

I wonder how modular property naturally arises in conformal theory. Is it obvious from physical viewpoint?
0
votes
1answer
193 views

About vertex algebra ,mode expansion

A vertex operator is a linear map associating every state to a operator-valued distributions(quantum field) on a algebra curve,which is also called operator-state correspondence. Chose a local complex ...
6
votes
4answers
1k views

Variants of Eisenstein irreducibility

In his article where he stated what we know as Eisenstein's irreducibility criterion (which actually was first proved by Schönemann, as was Scholz's reciprocity law and Hensel's Lemma), he ...
2
votes
1answer
883 views

Unique factorization in polynomial rings

Everybody knows that polynomial rings over fields have unique factorization, and that if $R$ has unique factorization, then so does $R[X]$. And everybody knows who proved these results first. Well, ...
9
votes
2answers
1k views

Galois group of a product of polynomials

How can I compute the Galois group of the polynomial $fg\in K[x]$ assuming that I know the Galois groups of $f\in K[x]$ and $g\in K[x]$? Let's suppose for simplicity that the field $K$ is perfect.
0
votes
2answers
231 views

Existence of an “anti-additive” (or “never linear”) map?

(I've edited this question) I'm searching for a continuously differentiable function $f:\mathbb R^2\to\mathbb R$ such that $f(x)+f(x+u+v)\neq f(x+u)+f(x+v)$ for all $x$ and all linearly independent ...
16
votes
4answers
2k views

Invariant Polynomials under a Group Action (hidden GIT)

Let's say I start with the polynomial ring in $n$ variables $R = \mathbb{Z}[x_1,...,x_n]$ (in the case at hand I had $\mathbb{C}$ in place of $\mathbb{Z}$). Now the symmetric group $\mathfrak{S}_n$ ...
33
votes
7answers
4k views

“Algebraic” topologies like the Zariski topology?

The fact that a commutative ring has a natural topological space associated with it is still a really interesting coincidence. The entire subject of Algebraic geometry is based on this simple fact. ...
3
votes
2answers
601 views

Injective modules and Pontrjagin duals

Forgive me for this naive question. We consider the following lemma and its proof in Lang's algebra, Third Ed., published 1999, Chap. 20, section 4, page 784. Every module is a submodule of an ...
4
votes
2answers
694 views

Is \mathbb{C}^{n \times n} algebraically closed?

Inspite of the fact that $C^{n \times n}$ is not a field, is it still possible to talk about it being 'algebraically closed' in the sense that $\forall f \in \mathbb{C}^{n \times n}[x]$ does $\exists ...
6
votes
2answers
478 views

Examples of Completions and Algebraic Closures

It is widely known that the algebaric closure of the $p$-adic completion $\mathbb{Q}_p$ of $\mathbb{Q}$ isn't complete anymore. It's completion is complete and known as $\mathbb{C}_p$. I have read ...
10
votes
3answers
873 views

Sign of infinite permutations?

Let $S_\infty$ the group of permutations of $\mathbb{N}$. It can be shown that there is no homomorphism $S_\infty \to \mathbf{Z}/2$ extending the sign on the finite symmetric groups. Is it possible to ...
16
votes
1answer
3k views

Infinite Tensor Products

Let $A$ be a commutative ring and $M_i, i \in I$ be a infinite family of $A$-modules. Define their tensor product $\bigotimes_{i \in I} M_i$ to be a representing object of the functor of multilinear ...
15
votes
4answers
6k views

Famous exercise from Lang's Algebra

There's a famous story about an exercise from Lang's Algebra that says something along the lines of "pick up a homological algebra book and prove all of the theorems yourself". I cannot find it in ...
4
votes
2answers
761 views

when are epimorphisms of algebraic objects surjective?

let $C$ be the category of $\tau$-algebras for some type $\tau$. consider the statements: every monomorphism is regular. every epimorphism in C is surjective. it is easy to see that 1. implies 2. ...
14
votes
2answers
3k views

Generalization of the shakehands/condom puzzle?

The classic handshake puzzle goes something like this: "Given that everyone has a different skin disease, how can you safely shake hands with 3 people when you have only 2 gloves?" Its common ...
36
votes
5answers
3k views

when is A isomorphic to A^3?

this is totally elementary, but I have no idea how to solve it: let $A$ be an abelian group such that $A$ is isomorphic to $A^3$. is then $A$ isomorphic to $A^2$? probably no, but how construct a ...
0
votes
2answers
3k views

Newbie boolean algebra question [closed]

The Majority Function is 1: A.B.¬C + A.¬B.C + ¬A.B.C + A.B.C I can see intuitively that it can be simplified to 2: A.B + A.C + B.C and thus A.(B + C) + B.C but how can I use boolean algebra to ...
3
votes
2answers
758 views

computation, algebra, logic

So a really simple way of describing a digital computer is to say that it is a device for performing boolean operations. You feed it a bunch of bit strings, which is a description of the problem and ...
2
votes
2answers
171 views

Decomposition result for multivariate polynomial

Let $k$ be a positive integer greater than $1$ and suppose that $F \in \mathbb{Z}[x_{1}, \ldots, x_{k}]$. Can we always find a natural number $n(k)$ and $f_{1}, \ldots f_{n(k)} \in \mathbb{Z}[x]$ ...
5
votes
3answers
223 views

Inverses in convolution algebras

Let $G$ be a locally compact totally disconnected group, and to make life easy let's suppose its Haar measure is bi-invariant. Let $C_c(G)$ be the space of locally constant complex functions on $G$ ...
14
votes
6answers
3k views

Do convolution and multiplication satisfy any nontrivial algebraic identities?

For (suitable) real- or complex-valued functions f and g on a (suitable) abelian group G, we have two bilinear operations: multiplication - (f.g)(x) = f(x)g(x), and convolution - (f*g)(x) = ...
32
votes
19answers
4k views

Canonical examples of algebraic structures

Please list some examples of common examples of algebraic structures. I was thinking answers of the following form. "When I read about a [insert structure here], I immediately think of [example]." ...