The abstract-algebra tag has no wiki summary.

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### Unique factorization domains

Let $c$ be an integer, not necessarily positive and $|c|$ not a square. Let $\mathbb{Z}[\sqrt{c}]$ be the set of complex numbers $$a+b\sqrt{c}, a, b\in \mathbb{Z},$$
which form a subring of the ring ...

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### Projectively splitting module

Is there a name for such class of modules $M$ such that $M\rightarrow N\rightarrow 0$ splits for every $N$ ?

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### Unit ideal in non-commutative rings [closed]

In a non-commutative ring (with identity), is it possible for an element which does not possess left or right inverses to generate the entire ring? i.e. $(r)=R$, where (r) is the two-sided ideal ...

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### Union of two proper subgroups [closed]

When can a group be written as the set-theoretic union of its proper subgroups?

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**1**answer

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### Bounds for zeros of polynomials with only real zeros

The formula in
http://de.wikipedia.org/wiki/Diskussion:Nullstelle#References
gives upper and lower bounds x1 and x2 for the roots of a polynomial all of whose roots are real.
Where can I find a ...

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### Does any textbook take this approach to the isomorphism theorems?

Below, I present an outline of a proof of the first isomorphism theorem for groups. This is how I usually think of the first isomorphism theorem for ______, but groups will get the points across. My ...

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**1**answer

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### What is the Schouten bracket for the Chevalley-Eilenberg complex with coefficients in a nontrivial module?

Let $\mathfrak g$ be a Lie algebra. The Chevalley-Eilenberg complex is defined to be $\wedge^* \mathfrak g$ with differential $d\colon \wedge^* \mathfrak g\to \wedge^{*-1}\mathfrak g$ defined by ...

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### Groups and rings which are not sets

An algebraic structure such as a group, ring, field, etc. is usually defined to be a set with some operations satisfying certain properties. I am curious what, if anything, goes wrong when the ...

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### Is Dependent Choice equivalent to the statement that every PID is factorial?

In this question, it was asked if AC is needed in the proof of the well-known fact that every principal ideal domain is factorial. As KConrad and Joel David Hamkins have pointed out, only DC, the ...

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### How do you decide whether a question in abstract algebra is worth studying?

Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my ...

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### How to tell if two random polynomials are identical

Let t be a positive real number. Let P(x) and Q(x) be two random polynomials with integer coefficients. If P(t) = Q(t), then what is the probability that P(x) is not identical to Q(x)?
Will it make a ...

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### Non-trivial consequences of Baer's theorem and Lucchini's theorem in subnormality theory

There are a couple of beautiful results in finite group theory that look trivial, at least on a first glance, but require non-trivial facts to prove. I am basically interested in whether these results ...

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### Admissible finite groups

I want to know when an abelian group of even order is admissible (or has a complete map)? And when a nonabelian group of even order is admissible (or has a complete map)?
Thanks.

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**1**answer

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### Rank of a linear combination of quadratic forms

Suppose we have a set of quadratic forms $Q_i (x_1, \dots, x_n)$ for $1 \leq i \leq k$ in $n$ variables, defined over $\mathbb{R}$. We suppose these are 'collectively nondegenerate' in the sense that ...

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### Consequences of not requiring ring homomorphisms to be unital?

As defined in many modern algebra books, a homomorphism of unital rings must preserve the unit elements: $f(1_R)=1_S$. But there has been a minority who do not require this, one prominent example ...

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### “Convex Optimization” over varying-dimension vector space?

For all instances of Convex Optimization I know of, the dimension of the vector space is defined beforehand.
Is there an area of mathematics that deals with "convex optimization" of varying-dimension ...

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### Sylow's theorem 3rd Proof Page 96 I.N.Herstein

I was just going through the 3rd Proof of Sylow's theorem given in the "Topics In Algebra" Book by I.N. Herstein. It looked very interesting and i really liked its Philosophy. My question what is its ...

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### Indeterminate “x” in Abstract Algebra/Ring Theory [closed]

How do you interpret the indeterminate "x" in ring theory from the set theory viewpoint? How do you write down R[x] as a set? Is it appropriate/correct to just say that
$R[x] = \{ f: R \to R | ...

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### Extension problem

As I understand, if $0\rightarrow A\rightarrow X\rightarrow B\rightarrow 0$ is a short exact sequence of abelian groups, $\mbox{Ext }_{\mathbb{Z}}^{1}(B,A)$ gives all the isomorphism classes of what ...

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### Linearly independent subsets of a free module

Is it true that the cardinality of every maximal linearly independent subset of a finitely generated free module $A^{n}$ is equal to $n$ (not just at most $n$, but in fact $n$)? Here $A$ is a nonzero ...

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### How fast are a ruler and compass?

This may be more of a recreational mathematics question than a research question, but I have wondered about it for a while. I hope it is not inappropriate for MO.
Consider the standard assumptions ...

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### Cardinality of symmetric group [duplicate]

Possible Duplicate:
Cardinality of the permutations of an infinite set
Why does the symmetric group on an infinite set X have the cardinality of the power set ${\cal P}(X)$?

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### Feit-Thompson Theorem: The Odd Order Paper

For reference, the Feit-Thompson Theorem states that every finite group of odd order is necessarily solvable. Equivalently, the theorem states that there exist no non-abelian finite simple groups of ...

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### Understanding the modules of semiprimitive rings

As far as I understand, a semiprimitive ring can be fully 'explored' by its simple modules, in the sense that a semiprimitive ring is the subdirect product of its simple modules (for brevity, I'll use ...

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### Explicit representations of finite fields

An old question that occurred to me again recently: are there any explicit formula known for sequences of irreducible polynomials $g_{p^n}(X)$ in $Z/pZ[X]$ such that for the finite field with $p^n$ ...

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### Is there a way to simplify block Cholesky decomposition If you already have decomposed the sub matrices along the leading diagonal?

Lets say we have a block matrix $ M =\left( \begin{array}{ccc}
A & B\\\\
B^{*} & C \end{array} \right)$ where M is positive definite. (A, and C are also pos def)
There is a formula for ...

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### Learning Algebra & Group Theory on my own [closed]

I'm learning Algebra & Group Theory, casually, on my own. Professionally, I'm a computer consultant, with a growing interest in the mathematical and theoretical aspects. I've been amazed with ...

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### The word “torsion” and its connection to geometry and homology

In an $R$-module $M$, an element $m \in M$ is said to be torsion if $am = 0$ for some $a \in R$ with $a \neq 0$.
Also, for a non-orientable (closed) surface such as the projective plane or the Klein ...

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### Does category theory help understanding abstract algebra?

I'm studying category theory now as a "scientific initiation" (a program in Brazil where you study some subjects not commonly seen by a undergrad), but as I've never studied abstract algebra before, ...

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### terminology about ring/algebra in abstract algebra and measure theory

Both in abstract algebra and measure theory is there term ring/algebra, but their definition are different and we can not deduce one from the other, the only requirement in definition they share is ...

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### What are the reasons for considering rings without identity?

I think a major reason is because Lie algebras don't have an identity, but I'm not really sure.

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### Free division rings?

Does it make sense to talk about, say, the free division ring on 2 generators? If so, does the free division ring on countably many generators embed into the free division ring on two generators?

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### Is the multiplication beetween even numbers an associative algebra?

We were discussing about the possibility of having an algebra over a field which is associative but has not the unity. Does it exist?
It has been proposed as a counterexample the set of even numbers. ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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.

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### 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, ...

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### 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 ...

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### 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} ...

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### 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?

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**1**answer

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### 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 ...

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### 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 ...

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### 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, ...

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### 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.

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### 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 ...

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### 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 $\mathfrak{S}_n$ ...

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### 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 ...

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### 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 ...