# Tagged Questions

53 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 ...
149 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 ...
24 views

### About product order [on hold]

Are there any references talking about product order on this wikipeida link? thanks!
44 views

### can somebody provide me this paper by NIVAN? [on hold]

equations in quaternions by I. Nivan published in AMM Vol 48 , 654-661 (1941). please i need it, and could not find it online anywhere.
34 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 ...
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 ...
36 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 ...
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 ...
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 ...
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 ? ...
62 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 ...
62 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 ...
73 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 ...
53 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 ...
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 ...
76 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 ...
27 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 ...
79 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 ...
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.
39 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: ...
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 ...
66 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?!
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 ...
140 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 ...
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. ...
74 views

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 ...
58 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 ...
110 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 ...
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 ...
42 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 ...
63 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 ...
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 ...
57 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 ...
60 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 $Algebra$ ...
42 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 ...
78 views

37 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 ...
228 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, ...
169 views

31 views

### Graph algebra papers

The following graph multiplication appears to be quite natural: Let $g_1=(V_1,E_1)$ and $g_2=(V_2,E_2)$ be two graphs ($V_i$ are sets of vertexes and $E_i$, sets of edges). Intuitively, the product I ...
216 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 ...
77 views

### Relational algebraic structures

Recently I came across the notion of relational $\beta$-algebra, defined as a set $S$ and a binary relation $\xi:\beta S-S$, where $\beta S$ denotes the set of ultrafilters on $S$ (and $\beta$ is the ...
### reference that any two elements of $S_n$ are conjugate iff they have the same cycle structure [duplicate]
How can I show/find a reference that any two elements of $S_n$ are conjugate iff they have the same cycle structure.