Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, and other advanced topics.

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Determining If Something Is A Set Is A Ring.

I had a question that I don't understand. I was looking through my abstract algebra textbook and I did a couple problem and I got most of the the problems right except for this one: $$R = {(a + ...
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130 views

Bachelor Thesis - Galois Theory Research Topics?

I'm on the last semester of my bachelor's degree (undergrad degree) and I will be writing my thesis next semester. I have talked to a professor at my university and one of the topics he suggested was ...
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Irreducibility of $x^p - y^q$ in $K[[x,y]]$, for p,q>1 relatively prime

For $p,q>1$, relatively prime, $x^p - y^q$ is irreducible in $K[x,y]$. Is it also irreducible in $K[[x,y]]$ and how would you show it? I'm quite stuck at the moment. Also $K[x,y]/(x^p - y^q)$ is ...
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51 views

Clarifying Semi-Direct Products: Example

I'm working through some questions on semi-direct products, and although I can work out these problems (for the most part), I usually have trouble completing them. I have identified some of the things ...
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64 views

Isomorphism of K-algebras

Let $k[x]$ be a polynomial ring and $I$ an ideal (resp. graded). If $k\subset K$ is a field extension, then prove that there is a natural (resp. graded) isomorphism of $K$-algebras: $$K[x]/IK[x]\cong ...
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46 views

Highest weights of irreducible components of tensor product of irreducible sl(3)-module.

I am study the representation theory of $sl(3)$ and I have a question about the tensor representation of irreducible $sl(3)$-modules as follows: For each weight $\mu$, let $L(\mu)$ be the irreducible ...
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51 views

Deducing that polynomials span

Let us say that we are dealing with a countable family of polynomials with real coefficients in $n$ indeterminates that commute. Are there any known/common nice systematic ways to tell if their span ...
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82 views

Application of Sylow Theorems.

I have a group theory exam coming up so I was reviewing through my textbook and I stumbled across a problem that looked interesting: Let $p < q$ be two primes and $G$ be a group of order $pq^3$. ...
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92 views

Ultraproduct of the algebraic closure of finite fields

Ok, I have done a little research on the next problem, and there are simple proofs of it using model theory, logic and the Łoś's theorem. But here, the idea is to prove it using only Field Theory, ...
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32 views

Correspondence between unipotent and nilpotent elements

Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. Let $\mathcal{U}(G)$ be the closed subvariety of unipotent elements of $G$, i.e., all elements whose ...
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125 views

Every finite map is surjective

I'm trying to understand the proof of the theorem which states that every finite map is surjective in Shafarevich's book: I didn't understand why the part underlined in red is true. I need a ...
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37 views

Global dimension of Artin algebras over a perfect field

Let $A$ be an Artin algebra over a perfect field $k$. Suppose that the global dimension of $A$ is finite. How one can prove that $$ \operatorname{gl}(A)=\min\{i \geq 0\mid ...
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71 views

Need hints on the following algebra problems.

I've been looking at these for over an hour and I don't understand how to do them. Any hints would be greatly appreciated. Let $p(x) = x^3 + x + 1$ and $F = Z_3[x]/\langle p(x)\rangle$. Factor ...
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36 views

Calculating a particular direct limit

Suppose we want to compute the direct limit of the direct system $$\mathbb{Z}^{14} \overset{A}{\longrightarrow}\mathbb{Z}^{14} \overset{A}{\longrightarrow}\mathbb{Z}^{14} ...
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47 views

Computing the fundamental groups of simple algebraic groups of type $A$

I'm interested in seeing the computation for the fundamental groups of the simple algebraic groups of type $A$. Below is the definition of the fundamental group for a simple algebraic group $G$. Let ...
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69 views

Parametric 12-deg and 14-deg equations with group $PGL(2,11)$ and $PGL(2,13)$?

We have, $$x^{12} - a x^{11} - 33x^8 + 22a x^7 - 11a^2 x^6 + 363x^4 - 121a x^3 + 121a^2x^2 - 23a^3x - 11^3 + a^4=0$$ $$x^{12} - a x^{11} - 11a x^9 - 44a x^7 - 88a x^5 - 88a x^3 - 32a x - a^2=0$$ ...
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116 views

Number of solutions to $x^2+y^2=1$ in a finite field?

This question is related to a simple case of a previous question: How many solutions are there for $$ x^2+y^2=1 $$ in a finite field $F_q$? The answer of the this question would give the ...
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62 views

Factoring irreducible polynomial over normal extension

Let $f$ be an irreducible polynomial over $F$ and $K/F$ be a normal extension. How to prove $f$ is factored by product of irreducible poly. over $K$ with same degree? I tried to do it by if $f_1, ...
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93 views

Decompose $X^{5}-2X^{4}+X^{3}-2X^{2}-2$ in $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$

Decompose $X^{5}-2X^{4}+X^{3}-2X^{2}-2$ in $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$ So far i've tried to use Eisenstein which failed, and i tried to decompose this with modular groups. With mod(2) we ...
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39 views

Equations with operations as variables

There are number puzzles which go like this: given 2_2_1=5. Insert operations (addition and multiplication) to make the equation valid. Solution: (+,+) or ($\cdot$,+). My question is: does anybody ...
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47 views

When does $\hom(A,-)$ have trivial kernel?

If $R$ is a commutative ring, is there a special name for those $R$-modules $A$ with the following property? $\forall R$-modules $M$: $~\hom(A,M) = 0 \Rightarrow M = 0$ Notice that every every ...
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63 views

Showing $|K : H \cap K| \le |G:H|$ where $H,K \le G$

I'd appreciate input on the validity or lack thereof of my attempted proof of the following: Let $H,K \le G$ be subgroups, where $|G:H| < \infty$. a) Show that $|K:H \cap K| \le |G:H|$. b) Show ...
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$\text{Hom}$ of irreducible modules and restrictions

This question is in reference to this paper. More specifically it is in reference to the proof of proposition 1.4 on page 8. First a defintion: Let $A$ be a semisimple finite dimensional ...
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45 views

Projective Module as a Direct Sum of Left Ideals

I wonder if the following statement is true: Every projective $R$-module is a direct sum of projective left ideals of $R$. Most examples of non-free projective modules I have seen are all left ...
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186 views

Does there exist a subring $R$ of $\mathbb{C}[x]$ such that $A \subsetneq R \subsetneq B$

I was thinking about this problem (all subrings here will refer to subrings that contain $\mathbb{C}$): Let $A,B$ be two subrings of $\mathbb{C}[x]$ such that ...
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103 views

Simplifying Relations in a Group

Let $K$ be the group generated by four elements $x_1,\cdots,x_4$ with relations that any simple commutator with repeated generator is trivial; for example, $[[x_2,[x_1,x_3]],x_3]=1$. As I have asked ...
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58 views

$R$ : a ring with unity, $a^2x=a$. Show that $ax=xa$.

Let $R$ be a ring with unity. For eah $a \in R$, there exists $x \in R$ such that $a^2x=a$. Show that $ax=xa$. I know that $R$ has no nonzero nilpotent elements and $axa=a$. Thus I tried to show ...
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140 views

How “bad” can presentation of the trivial group get?

These questions are sort of preliminary questions and reference requests for a project I am doing. Lets say, for concreteness, that $R$ is a set of words in the free group of rank two and that ...
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Proof that $\mathbb Z[\sqrt{3}]$ is Euclidean

Let $R_d$ be the ring defined as $R_d=\left \{ x+y\omega : x,y\in \mathbb{Z} \right\}$, where $$\omega = \begin{cases} \sqrt{d}, & \text{if } \quad d \not \equiv 1\mod 4 \\ \frac{1+\sqrt{d}}{2}, ...
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Find the fundamental group and the Alexander polynomial

I would like to find the Alexander polynomial of the link $L$, described below. Let $K(q,r)$ be the $(q,r)$-torus knot embedded on a torus $V$. Inside the torus $V$, consider a smaller solid torus ...
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Are there some kind of “multialgebras” with terms or equations, where an operation can result with different values in different places?

Many-valued (multivalent, polivalent) operations are studied in multialgebras. Applied to a certain value of its argument, a many-valued operation o(x) can result in different values. But in ...
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Finitely generated group has only finitely many subgroups of given index

In my previous question i have got a comment that : If a group is finitely generated, then there are finitely many subgroups of a given finite index. I do not yet see the beauty of this problem ...
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$N\unlhd G$ such that $|N|<\infty$ then we have $M\unlhd G$ with $|G/M|<\infty$ and $mn=nm ~\forall m\in M,n\in N$

Now, the question is : Assuming $N\unlhd G$ such that $|N|<\infty$ Question is to prove that : we have $M\unlhd G$ with $|G/M|<\infty$ and $mn=nm ~\forall m\in M,n\in N$ As i am assuming $G$ ...
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Origin of the terminology “connected algebra”

I was wondering what is the origin of the word "connected" for a connected algebra ? To be more precise, why is a graded $R$-algebra $A_{\ast}$ with an augmentation $A_{\ast} \to R$ that restricts to ...
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On the definition of Fox derivative

I am reading An Introduction to Knot Theory by W.B. Raymond Lickorish. In Chapter 11 the motivation for the Fox derivative is mentioned. I understand why the contribution of the occurrence of $x_j$ in ...
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81 views

Distributive Property - A Converse Theorem

Suppose that $\ast$ distributes over $+$, where $0$ is the additive identity. We can conclude the following. $a \ast b = a \ast \left( b + 0\right)$ $a \ast b = a \ast b + a \ast 0$ $\therefore a ...
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Trying to prove structure result for ${\rm Hom}(A,B)$

Let $A$ be a $\mathbb Z$-algebra that is finitely-generated and free as a $\mathbb Z$-module and let $\pi: A \rightarrow \mathbb Z$ a nontrivial $\mathbb Z$-module homomorphism. For a positive ...
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Free Graded Commutative Algebra on a Graded Vector Space

Let $V$ be a graded vector space, thought of as a collection $\{ V^n \}_{n \ge 0}$ of vector spaces. Let $V_{odd} = \bigoplus_{n \text{ odd}} V^n$ and $V_{even} = \bigoplus_{n \text{ even}} V^n$. I am ...
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Fixed points of group actions in convex symmetric sets

Let $X$ be a real vector space and $G$ be a group. Suppose that $G$ acts linearly on $X$, in the sense that for any $\alpha,\beta\in\left[0,1\right]$ , $g\in G$ and $x,y\in X$, we have $$g(\alpha ...
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free dimension of a module?

For any module $M$ there is defined the projective/injective/flat dimension, which is the length of the shortest projective/injective/flat resolution of $M$. Why isn't there defined a free dimension ...
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Equivalence classes of triplets satisfying $x^2+y^2+z^2=0$ over $\mathbb{F}_p$

The affirmative answer to this question illustrates that the equation $$x^2+y^2+z^2=0$$ has $p^2-1$ nontrivial solutions over $\mathbb{F}_p$ (solutions that are not $(0,0,0)$). If $(x,y,z)$ is a ...
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How can we define “trivially orthogonal” groups?

In any lattice-ordered group, we say that two elements are orthogonal if their meet is 1. I've been thinking of groups who have only "trivial" orthogonal relations, i.e. $x\perp y\implies x=1$ or ...
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86 views

Shortest way to convert two formulas using associative and commutative steps

You have two formulas in which you add $n$ variables. The variables in the two formulas are the same, but they may be in a different order and the parenthesis may be different. As an example I will ...
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Help with a proof of Sharp's Steps in commutative algebra

I'm trying to understand the following part of this book I couldn't prove that $aR=\Pi_{i=1}^sRp^{t_i}_i$. $\subset$ part: $x\in aR\implies x=ar_1\implies x=up_1^{t_1}...p_s^{t_s}r_1\implies ...
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81 views

When powers of matrices are represented as a sum of integral matrices

There is given a ring $R$ and a subring $K$ with unit. We have a matrix $A$ of size $n$ over $R$. The characteristic of $R$ is $0$ or more than $n$. The statement is: If $A^m$ for some ...
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meaning of ``partial converse''

In the definition of a commutative ring $(R,+,\times)$, one of the postulates given is that of uniqueness, i.e. that $$ a=a', b=b'\implies a+b=a'+b', ab=a' b'.$$ The text states that for the system ...
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167 views

Recognizing subadditivity

Let $f: (0,\infty) \to \mathbb{R}$ be some continuous function. We say that $f$ is subadditive if the bound \begin{align} f(x+y) \leq f(x) + f(y) \tag{1} \end{align} holds. I was attempting to ...
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Cartan's Criterion. $L$ solvable $\implies$ $Tr(xy) = 0$ $\forall x \in L$, $\forall x \in L^{(1)}$

Cartan's Criterion. Given $V$ a finite dimensional complex vector space and $L$ a Lie subalgebra of $gl(V)$ then, $L$ solvable $\implies$ $Tr(xy) = 0$ $\forall x \in L$, $\forall x \in L^{(1)}$. ...
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53 views

Can all non-archimedean groups be written as a product of archimedean groups?

All the non-archimedean groups I know of can be written as the product of archimedean groups. I'm wondering if this is generally true. I think I've found a proof, but I haven't heard this theorem ...
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129 views

mapping cones of chain homotopic maps

Suppose that $ f $ and $ f' : C \to D $ are morphisms of chain complexes; Cone($f$) is the mapping cone of $f$; if $f$ and $f'$ are chain homotopic, what is the relation between Cone($f$) and ...