1
vote
1answer
23 views

Characterization of solvable groups in terms of subgroups of certain orders?

In this question, the OP mentions the following result: a finite group $G$ is solvable if and only if $$\text{for all $n$ dividing $|G|$ such that $\gcd(\frac{|G|}{n},n)=1$, $G$ has a subroup order ...
3
votes
0answers
37 views

Best algebra text for Model Theory

I'm looking for an algebra book that is tailored towards some of the ideas in Model Theory, I'm currently slogging through Hodges' Model Theory. I'm a bit rusty with my algebra and was curious if ...
1
vote
1answer
29 views

Localization of the Integer Ring

Let $\mathbb{Z}$ be the ring of integers and let $p$ be a prime, then the $p$-localization of $\mathbb{Z}$ is defined as $\mathbb{Z}_{(p)}=\{\displaystyle\frac{a}{b}|a,b\in\mathbb{Z},p\nmid b\}$. I ...
4
votes
2answers
55 views

module over a quotient of a principal ideal domain

The Statement I suspect the following proposition is well known, but I found no reference. Proposition If $A$ is a principal ideal domain, if $I$ is a nonzero ideal of $A$, and if $M$ is an ...
1
vote
1answer
30 views

Lie ideals of $gl_n(K)$

I am looking for some reference where I can find a detailed study of the Lie ideals of the general linear Lie algebra $gl_n(K)$ with the bracket $[A,B]=AB-BA$, where $K$ is a field (if there are ...
9
votes
5answers
333 views

Exercises in category theory for a non-working mathematican (undergrad)

I'm trying to learn category theory pretty much on my own (with some help from a professor). My main information source is the good old Categories for the working mathematician by Mac Lane. I find the ...
2
votes
1answer
45 views

Is there something that studies equivalent forms of writing and expression?

Supose we have: $x^2+x$, one could write it as $x(x+1)$ which would be equivalent to the first expression. I guess there might be a finite number of ways of writing expressions such that they are ...
6
votes
0answers
110 views

Category Theory textbook (learning through guided discovery Dummit and Foote)

sorry for asking the same question in a slightly different angle, I want a book in Category Theory similar to Dummit and Foote's book in Abstract Algebra. I want it to have tons of examples and ...
14
votes
2answers
253 views

Quotient of polynomials, PID but not Euclidean domain?

While trying to look up examples of PIDs that are not Euclidean domains, I found a statement (without reference) on the Euclidean domain page of Wikipedia that $$\mathbb{R}[X,Y]/(X^2+Y^2+1)$$ is ...
-1
votes
0answers
37 views

Behavior of groups under extension

We know that an extension of a solvable group by a solvable group is solvable. Similarly we can find other properties of group extensions here Can someone provide a reference to these statements where ...
1
vote
1answer
46 views

When is a binary operation bipotent?

I learnt that $\max(-,-)$ is a bipotent binary operation but I'm not able to find a definition of bipotent operation. QUESTION A binary operation $*:M\times M \rightarrow M$ is bipotent if ...
0
votes
1answer
39 views

Lie rings: reference request

Dear friends: I am looking for a modern reference for Lie rings (In particular, I would like to have nice references for the structure of Lie ideals), let it be lecture notes or a book, in the sense ...
1
vote
2answers
47 views

roots of cubic polynomial

On page 26 of Milne's Elliptic Curves (http://www.jmilne.org/math/Books/ectext5.pdf), he states the following: "... a cubic polynomial $h(x) \in k[x]$ with two roots in $k$ has all of its roots in ...
8
votes
3answers
1k views

Is it bad to keep aside Lang's Algebra in graduate school?

Question is as it is stated in title. I will be joining for PhD program in this July 2014. I am interested in working in Algebra/Algebraic Geometry/Algebraic Number Theory. I tried to learn algebra ...
0
votes
0answers
22 views

Submodule of a free module over a PID with infinite rank [duplicate]

Let $R$ be a PID. We know that a submodule of a free $R$-module with rank $n < +\infty $ is free with rank $ \leq n $. But if $M$ is a free $R$-module of infinte rank does this fact remain true ? ...
1
vote
1answer
67 views

Primary decomposition of $I = (x^2, y^2, xy)$

I want to find a primary decomposition of the ideal $$ I = (x^2,y^2,xy) \subset k[x,y]$$ where $k$ is a field. How to proceed? Are there algorithms to find such decompositions? Where can I find ...
1
vote
1answer
66 views

Locally finite infinite field

Is there a place (a book maybe) where I can find some useful information on infinite locally finite fields? Especially when all of whose proper subfields are finite? I know, for instance, that a ...
6
votes
1answer
77 views

Follow up to Pinter's abstract algebra

I wanted to learn abstract algebra this summer so I bought Pinter's A book of Abstract Algebra. I was planning on reading it over the course of the summer, but just finished the last problem of its ...
2
votes
1answer
59 views

Solving polynomial equations over finite fields

I have looked (a bit) at questions like finding the number of roots of $x^n =1$ over a finite field. Now I would like to understand how to solve polynomial equations over finite fields. From what I ...
0
votes
0answers
22 views

Endomorphisms of Groups - Book Recommendation

Which books dealing with group theory have considerable material on endomorphisms? The books I have seen usually have something on homomorphisms, isomorphisms, and automorphisms, but very little on ...
2
votes
3answers
79 views

Reference request: algebraic methods in geometry

I am a (soon to be) third year undergraduate who has just finished courses in linear and abstract algebra. While I enjoyed the study of algebraic structures in their own right, my favorite part of the ...
2
votes
0answers
33 views

Tiling a rectangle and tensor products

Consider the following theorem: Whenever a rectangle is tiled by rectangles each of which has at least one integer side, then the tiled rectangle has at least one integer side. There is a paper ...
4
votes
2answers
80 views

What is a “connection” in algebraic terms?

It seems that I read this somewhere else, but I did not find the correct reference now. We know that a vector bundle $E\to M$ is a (projective or locally free) module of $C^\infty(M)$. Then how to ...
1
vote
0answers
142 views

Generalization of Chinese Remainder Theorem to infinite ideals

I'm looking for any (obviously weaker) generalization of this famous theorem in the special case that the family of ideals is not finite.
0
votes
0answers
40 views

Request for three foundational papers by Oystein Ore.

Does anybody know how to get or how can I find the following foundational papers on (permutable) groups: O. Ore: Contributions to the theory of groups of finite order (ProjectEuclid Link) O. Ore: ...
1
vote
0answers
54 views

Does This Ring have a Name?

Let $M_1=\{0,1,2,4,5,8,9,10,\cdots\}$ be the set of nonnegative integers that can be written as a sum of two perfect squares. Let $M_2=\{\sqrt{m}: m\in M_1\}=\{0,1,\sqrt{2},2,\sqrt{5},\cdots\}$. Let ...
1
vote
1answer
67 views

Math competitions resource at university level

I want some problems especially in Algebra field for math competitions at undergraduate math students level. Does anybody here know book, website,... that I can use?!
3
votes
0answers
38 views

Correspondence theorem for rings.

Could someone provide a reference that includes a full and honest proof of the Correspondence Theorem for rings? Let $A$ be a multiplicative ring with identity and $I$ an ideal of $A$. There is a ...
5
votes
3answers
147 views

Open Source Abstract Algebra Textbooks

Does anyone know of any open source abstract algebra textbooks other than Judson's? I am about to write a small program for a friend that will generate a random algebra problem (for preparing for ...
1
vote
0answers
46 views

Branch of mathematics that studies groups / rings or rational functions

I'm not really a mathematician, and looking for some literature which could potentially help me in research. Im interested in algebra of rational functions (ratios of polynomials) of finite order. ...
0
votes
0answers
74 views

Readings for Noether

I'm studying the theory of Noether but I have only 4 pages of lecture notes with no details or examples. Are there any good lecture notes or chapters you know about? In my lectures the basics of ...
1
vote
1answer
59 views

A theorem about ideals of $K[T_1,\ldots,T_n]$ and their generators

Suppose that $L\subseteq K$ is a field extension ( we are in characteristic $0$) and moreover that $\mathfrak a\subseteq K[T_1,\ldots,T_n]$ is an ideal ($T_1,\ldots,T_n$ are indeterminates). I have ...
6
votes
2answers
115 views

What is $\operatorname{Hom}_\mathbb{Z}(\mathbb{Q}/\mathbb{Z},\mathbb{Q}/\mathbb{Z})$?

Is $\operatorname{Hom}_\mathbb{Z}(\mathbb{Q}/\mathbb{Z},\mathbb{Q}/\mathbb{Z})$ isomorphic to any "known" group? I suppose what I mean is, is it isomorphic to a group that isn't a Hom group? If such ...
1
vote
0answers
33 views

When is the dot product of roots of certain multivariate polynomials also a root?

Problem. Let $n\in\mathbb N$ be fixed, and suppose that we are given three collections $$z_1,\ldots,z_n\in\mathbb Z,~a_1,\ldots,a_n\in\mathbb R,\text{ and }b_1,\ldots,b_n\in\mathbb R.$$ Suppose we ...
1
vote
0answers
43 views

Errata for Vinberg's “a course in algebra”?

I'm reading Vinberg's "a course in algebra" to have a basic understanding. I wonder if there is an errata for the book? For example, Example 1.52 on page 23 talks about "$x^2+x+1=0$" , however I ...
1
vote
2answers
64 views

Introductions to posets on algerbaic structures (Everything I need to know about them)

I need a good and complete introduction to Tree-like orders and partial orders on algebraic structures with one operations. I accept basic texts too. I'm looking for free online texts mostly because ...
2
votes
0answers
33 views

Fractional linear transformations and the extended complex plane in a more abstract context?

Does anyone know of an "abstract algebra-esque" treatment of the extended complex plane and the Mobius transformations? I am studying complex analysis now, and I am a little frustrated that my ...
5
votes
0answers
59 views

Literature to the ring $\mathbb{Z}[\phi]$ where $\phi=\frac{1+\sqrt{5}}{2}$ is the golden ratio

I know few about algebraic number theory but recently I stumbled upon the ring $\mathbb{Z}[\phi]$ where $\phi = \frac{1+\sqrt{5}}{2}$ is the golden ratio. It seems to be a very interesting object to ...
3
votes
1answer
61 views

Books on Rings without Identity

I was just wondering if anybody knows of any good books or articles that study rings (and algebras) without (or not necessarily with) identity. I have gone through Thomas Hungerford's $Algebra$ ...
2
votes
1answer
43 views

Is every element of a complex semisimple Lie algebra a commutator?

Let $L$ be a (finite-dimensional) complex semisimple Lie algebra. Then we know that $L = [L,L]$. Is it true that every element of $L$ must be a commutator? Since a complex semisimple Lie algebra is ...
0
votes
1answer
78 views

$\biggl( \prod_{i \in \mathbb{N}}\mathbb{Z}/p^i\mathbb{Z}\biggr)\otimes_{\mathbb{Z}} \mathbb{Q}\ne 0$

I've found this claim $$\biggl( \prod_{i \in \mathbb{N}}\mathbb{Z}/p^i\mathbb{Z}\biggr)\otimes_{\mathbb{Z}} \mathbb{Q} \not\cong \prod_{i \in \mathbb{N}}\biggl( ...
2
votes
0answers
59 views

Question about definition of some classes of bimodules.

Suppose that we have a ring $R$ and a $R$-$R$ bimodule $M$ such that: For every $r\in R$ and $m\in M$ there exists $r'\in R$ such that $m\cdot r=r'\bullet m.$ Examples of this bimodules can be seen ...
10
votes
4answers
294 views

The automorphism group of the real line with standard topology

How much is known about the automorphism group of the real line with the standard topology? I have been unable to find a reference for this question. Any information about $\mathrm{Aut}(\mathbb R)$ or ...
1
vote
1answer
37 views

What are the roots of unity quadratic integers?

The article of Roots of unity in wikipedia implies that the following roots of unity are quadratic numbers: $$ \{\pm 1\}, \{\pm 1,\pm i\}, \{\pm 1,\pm \zeta,\pm \zeta^2\}. $$ where $\zeta=\exp(2\pi ...
4
votes
2answers
81 views

Primitive elements for $K=\Bbb{Q}(\sqrt{2},\sqrt{3})$

The key lemma for proving the primitive Element Theorem (for finite extension of a field $F$ with characteristic $0$) in Artin's Algebra (2nd edition) is the following: Suppose $char F=0$ and ...
2
votes
1answer
64 views

What sort of algebra is this?

Let us say that I have a set of symbols, $S$. The symbols can be operated on by a set of $n$-ary operators, $O$. Importantly, some of these operators are in the set of symbols, i.e. $S \cap O \neq ...
2
votes
1answer
38 views

Arrow's Impossibility Theorem Using Boolean Algebra

I am currently working on a research project which involves using Boolean matrices for the proof of Arrow's Impossibility Theorem and various other lemmas and results related to quasi ordered sets. In ...
11
votes
2answers
230 views

Putting down axioms for some symbols. Playing with their consequences qualitatively and symbolically. Building theories. The book?

I am interested in the design and building of theories. By building theories, I mean putting down axioms of various kinds, over various fields, exploring their perhaps interesting, or probably boring, ...
4
votes
0answers
173 views

Conjugation Quandles and… “Quandle-Groups”? From quandles to Groups.

A quandle $(Q,*,/ )$ is a idempotent right-distributive and right invertible structure. 1) $a*a=a$ 2) $(a*b)*c=(a*c)*(b*c)$ 3) $(a*b) /b=(a/b)*b=a$ If we have a group $(G, \cdot, ...
1
vote
0answers
41 views

coalgebra/algebra of the identity endfunctor

Let $\mathbb{C}$ be a small/locally small category and let $T:\mathbb{C} \to \mathbb{C}$ be an endofunctor. One can then have $T$-algebras and $T$-coalgebras in the usual way: for $X,Y \in ...