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

learn more… | top users | synonyms (1)

1
vote
2answers
52 views

question asked around a weird concept

I am struggling with these questions. I dont know what is meant by the term system of representatives. anybody knows about these things?
-2
votes
0answers
11 views

Forming a Cayley table [on hold]

How to form a Cayley table using $*$ as a binary operation on $P(A)$, where $A=\{1,2,3\}$. And the solution of main diagonal in the Cayley table.
0
votes
1answer
25 views

For $N, M \unlhd G$ relation between $MN/(M\cap N)$ and $N/(M\cap N)\times M/(M\cap N)$

Let $N, M \unlhd G$. Is $MN/(N\cap M)$ isomorphic to some subgroup of $$ N/(N\cap M) \times M/(N\cap M) $$ and how to prove this?
2
votes
1answer
23 views

A surjective endomorphism (of a Noetherian ring) is injective.

The problem is stated as follows: "Let $R$ be a Noetherian ring and $\theta$ be a ring homomorphism from $R$ to $R$. Show that if $\theta$ is surjective then it is also injective." Regardless of the ...
-3
votes
1answer
25 views

Distinct elements of relations

R is the relation on Z (integers) given by xRy ( X is related to Y ) if 3 divides (x-y), what are the distinct elements of R?
4
votes
3answers
151 views

Is $\sqrt[3]{-1}=-1$?

I observe that if we claim that $\sqrt[3]{-1}=-1$, we reach a contradiction. Let's, indeed, suppose that $\sqrt[3]{-1}=-1$. Then, since the properties of powers are preserved, we have: ...
0
votes
1answer
30 views

group theory dihedral group problem [on hold]

I am stuck in this problem. plz give some suggestion
5
votes
0answers
82 views

Diophantine equation: $13^x+3=y^2$

$$13^x+3=y^2\iff \left(4+\sqrt{3}\right)^x\left(4-\sqrt{3}\right)^x=\left(y+\sqrt3\right)\left(y-\sqrt3\right)$$ $\gcd\left(y+\sqrt3, y-\sqrt3\right)=1$, therefore ...
-3
votes
1answer
31 views

Proving that R is a equivalence relation. [on hold]

Let $R$ be a relation on $\mathbb Z$ given by $xRy$ iff $3|(x-y)$. How do we prove that R is an equivalence relation?
0
votes
1answer
31 views

The Binary Operation [on hold]

Let $A$ is equal to $\{ 1,2,3 \}$. Binary Operator defined as $\star$ on $\mathcal P (A)$ by $X\star Y$ is equal to $(X-Y) \cup (Y-X)$ and also equal to $X\triangle Y$, where: $X,Y$ is a subset of ...
7
votes
0answers
29 views

Number of elements of the group $SL_6(\mathbb{Z}/p^k\mathbb{Z})$

For $p$ a prime, what is the number of elements of the group $SL_6(\mathbb{Z}/p^k\mathbb{Z})$, $k \ge 1$? I can answer the $k=1$ case. For each element of $SL_6(\mathbb{Z}/p\mathbb{Z})$, there are ...
0
votes
0answers
35 views

a problem in homological algebra

For $C$ is abelian group satisfied $pC=0$ with p is a prime number and $G$ is abelian group. prove that $Ext_{Z}(C,G)\cong Hom(C,G/pG)$ I thought about this in 2 hours but couldn't prove it!
1
vote
0answers
102 views

Maps to quotient rings

If $R$ is a ring, $\mathfrak{a}$ is an ideal of $R$ and $S=A[x,y,z,\dots]$ where $A$ is a commutative ring, then is there a ring $S^\prime$ such that there is a bijection: ...
0
votes
1answer
58 views

Roots of $\Phi_{31}(x)$ as roots of unity.

Let $\Phi_{31}(x)$ be the $31$-cyclotomic polynomial. I want to show that $\Phi_{31}(x)$ is the product of six irreducible quintic factors in $\mathbb{F}_2$. I am running into difficulties ...
0
votes
1answer
66 views
+50

A question about an infinite sequence of elementary row operations

Do there exist matrices $A$ and $B$ such that $B$ can be transformed into $A$ only if an infinite number of elementary row operations are performed on $B$? "What can we multiply the top equation by ...
1
vote
0answers
23 views

Composition of module homomorphisms and their kernels

Let $R$ be a principal ideal ring and let $I$ and $J$ be two ideals of $R$. Suppose $\phi: R \times R \rightarrow R \times R$ and $\psi: R \times R \rightarrow R \times R$ are two $R$-module ...
0
votes
0answers
27 views

Kaplansky characterization of principal Artin ring

I would like to learn the proof of this paper of Kaplansky where it is proven that for a commutative ring every module split as sum of cyclic module iff the ring is an Artin principal ideal ring (well ...
0
votes
2answers
27 views

Inverse images of ideals

I was trying to solve the following exercise: Let $f\colon R\to S$ be a ring epimorphism, $I \subseteq S$ be an ideal, and $J = f^{-1}(I)$. Check that if $I$ is maximal (resp. prime) then $J$ is ...
0
votes
2answers
13 views

Equivalence classes of relation $\rho: (x,y)\in \rho \Leftrightarrow (\exists k \in \mathbb{Z})x-y=3k$

I don't understand how equivalence classes are $$C(1)=\{3k+1:k\in \mathbb{Z}\}$$ $$C(2)=\{3k+2:k\in \mathbb{Z}\}$$ $$C(3)=\{3k:k \in \mathbb{Z}\}$$ Could someone explain?
4
votes
0answers
27 views

Which of the algebra isomorphisms hold?

Fix $m, n \ge 1$. Which of the algebra isomorphisms below hold? $k\langle t_1, \dots, t_m\rangle \otimes_k k\langle s_1, \dots, s_n\rangle \cong k\langle t_1, \dots, t_m, s_1, \dots, s_n\rangle$ $k[ ...
2
votes
1answer
34 views

exercise Question-29 from contemporary abstract algebra [on hold]

Consider the element A=(1101) in SL(2,R) what is the order of A? If we view A=(1101) as a member of SL (2,Zp), what is the order of A
4
votes
1answer
39 views

Example of an nonidentity element in the kernel of the map.

This question is related to my previous question here. Let $n, m > 1$. The map $\mathbb{Z} \twoheadrightarrow \mathbb{Z}/m\mathbb{Z}$, of reduction mod $m$, induces a group homomorphism $F: ...
3
votes
2answers
29 views

Why do “the Dynkin components of a weight play the role of eigenvalues with respect to the generators $H^i$ of the Cartan subalgebra”?

In the book "Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists" by Jürgen Fuchs,Christoph Schweigert the authors write "In the description of representations, the ...
9
votes
1answer
59 views

Induced group homomorphism $\text{SL}_n(\mathbb{Z}) \twoheadrightarrow \text{SL}_n(\mathbb{Z}/m\mathbb{Z})$ surjective?

Let $n, m > 1$. The map $\mathbb{Z} \twoheadrightarrow \mathbb{Z}/m\mathbb{Z}$, of reduction mod $m$, induces a group homomorphism $F: \text{SL}_n(\mathbb{Z}) \to ...
1
vote
1answer
18 views

Is the fundamental weight basis (a.k.a Dynkin basis) an orthonormal basis?

The simple root $\alpha_i$ basis is not an orthonormal basis, as can be seen from the Cartan matrix, which encodes how much they aren't orthonormal. For simplicity, let's assume a simply-laced Lie ...
5
votes
0answers
33 views
+50

Closed formula for Poincaré series in terms of adjacency matrix.

Let $Q$ be a finite quiver with vertex set $I$. For each $n = 0, 1, 2, \dots,$ let $k^{(n)}Q \subset kQ$ be the $k$-linear span of all paths of length $n$, in particular, we have$$k^{(0)}Q = ...
0
votes
1answer
25 views

Degrees of freedom of a complex vector space $V$ and its conjugate $\bar V$?

As an easy example consider the complex vector space $\Bbb C^2$. We can consider $\Bbb C^2$ as vector space over $\Bbb R$ and thus have the four basis vectors $$ \hat e =\{(1,0), (i,0), (0,1), ...
4
votes
3answers
82 views

For a prime integer $p \in \{2, 3, 5, \cdots\}$, is $pR$ a maximal ideal in $R$?

If $R$ is a commutative ring with unit and $p$ is a prime number ($2,3,5,\cdots$), then is $pR$ a maximal ideal of $R$? If not what conditions should I impose on $R$?
0
votes
0answers
17 views

Automorphism of two members as Generator

Let $X=\langle a,b|a^{2^m}=b^{2^n}=1,[a,b]=a^{2^{m-1}}\rangle$, $m,n\ge 2$ If $\alpha \in Aut(X)$ (Automorphism Group of $X$) is defined as \begin{cases} \alpha(a)=a^{2^{m-1}+1}\\ ...
2
votes
1answer
99 views

How can I solve $x^2+2=y^3$ in $\mathbb{Z}$?

Prove that $\left \{ (x,y)\in\mathbb{Z}^2:x^2+2=y^3 \right \}\subseteq \left \{ (-5,3),(5,3) \right \}$.
3
votes
1answer
52 views

Prove that $f_n=-1+\prod_{i=1}^{n}(X-i)$ is irreducible in $\mathbb{Z}[X]$

Prove that, for all $n\in \mathbb{N}$, $f_n=-1+\prod_{i=1}^{n}(X-i)$ is irreducible in $\mathbb{Z}[X]$.
0
votes
1answer
52 views
+50

Proving the Ideal Generated by the Coefficients of $f(X)\cdot g(X)\in R[X]$ is $R$.

Let $R$ be a commutative ring with unity, and let $f(X),g(X)\in R[X]$. Assume the ideals generated by the coefficients of $f(X),g(X)$ are both $R$. Prove that the ideal generated by the ...
0
votes
0answers
7 views

Nakayama automorphism $\sigma$ of Hecke Algebra $^0H^f_n$ is not inner for $n\geq 3$?

With $R=\mathbb{Z}[q_1,q_2]$, the Hecke algebra $H^f_n$ of $S_n$ is defined to be the $R$-algebra generated by $T_1,\dots,T_{n-1}$ satisfying $T_iT_{i+1}T_i=T_{i+1}T_iT_{i+1}$, $T_iT_j=T_jT_i$ if ...
1
vote
0answers
32 views

Equivalence of two statements in a field.

I need to prove the following in order to prove something interesting about generalized quaternions: Let $K$ be a field and suppose $a \neq 0, b \neq 0$ are elements of $K$. Then the following are ...
2
votes
1answer
54 views

What is the smallest subfield of the complex numbers which has the property that every polynomial of odd degree has a root

It can be shown using the intermediate value theorem that every polynomial of odd degree with real coefficients must have at least one real root. I was just curious, are there any other smaller fields ...
1
vote
1answer
43 views

If $|xH|$ has order $n$, then there is an element $y$ with $|y|=n$ and $xH=yH$

Let $G$ be a group, and let $H$ be a normal subgroup with $|H|=m$. Suppose $n$ and $m$ are relatively prime. If $|xH|$ has order $n$, we wish to find an element $y$ with $|y|=n$ and $xH=yH$. It is ...
-1
votes
0answers
44 views

Show that $\langle x,y\rangle$ is not projective as a $\mathbb{Q}[x,y]$ -module. [closed]

I took this exercise for a long time but I can't prove it. Show that $\langle x,y\rangle$ is not projective as a $\mathbb{Q}[x,y]$ -module. Anyone could help me?
0
votes
0answers
32 views

Notation/definition problem for commutative binary operation

I'm trying to describe/define the commutative binary operation on a three-element set which when the operands are the same, gives the same element and when they are different gives the element which ...
0
votes
1answer
22 views

Minimal polynomial with repeated factors over an algebraically closed field.

Let $k$ be an algebraically closed field and let $V$ be a vector space over $k$ and let $T: V \to V$ be any linear transformation. I can't think of an example when the minimal polynomial of $T$ will ...
1
vote
1answer
59 views

If $[A,A]A[\lambda,A] = 0$ then $\lambda \in Z(A).$

Suppose that $A$ is a unital ring and $([A,A]) = A.$ If $[A,A]A[\lambda,A] = 0$ prove that $\lambda \in Z(A).$ Comments: This is part of an exercise I'm doing, I'm posting this part because I am ...
4
votes
4answers
70 views

Let $G$ be a compact group. If $\{a^n\}_{n \in \mathbb{Z}}$ is dense in $G$, then $G$ is abelian.

It was used in the middle of a theorem's proof and I am not sure how to prove this fact.
1
vote
2answers
38 views

Roots of unity over a general field $k$.

Let $k$ be an algebraically closed field of characteristic zero. Then $x^4-1$ factors linearly in $k$. Usually, I would consider the roots of $x^4-1=0$, the fourth roots of unity. Which in the case $k ...
0
votes
0answers
31 views

Show that the homomorph image of an abelian group is abelian

Since $G$ is abelian, we have that: $$ab = ba \implies \phi(ab) = \phi(ba) \implies \phi(a)\phi(b) = \phi(b)\phi(a)$$ Am I rigth?
1
vote
1answer
51 views

Embedding tensor product of integral domains

Let $C$ be a subring of integral domains $A,B$ and let $C',A',B'$ denote their field of fractions respectively. Can we always embed $A\otimes_CB$ in $A'\otimes_{C'}B'$ by $a\otimes b\mapsto ...
0
votes
1answer
48 views

Determine number of elements of order 12 of a group

Let's say we have a commutative group G that's specified by generators and relations. We find that the group G normal form is: $Z_2\times Z_6\times Z_{12}$ and that the elementary form is $Z_2\times ...
0
votes
0answers
25 views

Roots of unity, intersection of fields [duplicate]

How to prove that intersection of $\Bbb{Q}(m)$ and $\Bbb{Q}(n)$ is equal to $\Bbb{Q}$, where $(m,n)=1$ and $\Bbb{Q}(n)$ is $n$-th cyclotomic field?
1
vote
1answer
70 views

Multiplication without exponents

Edited for clarity (The original statement of the problem I was explaining the problem as a redefinition of multiplication, as can see from the original comments; but I think the algorithmic ...
1
vote
0answers
37 views

Primitive vectors in $A^n$

Let $A$ be a commutative ring with 1. Let n be a positive integer. I call a vector $(a_1,...,a_n) \in A^n$ primitive, if the ideal generated by $\{a_1,....,a_n\}$ is $A$. Question: Given a primitive ...
0
votes
2answers
63 views

If $\phi$ is an isomorphism, $\phi(g)^n = 1 \iff g^n = 1$. Doesn't this hold for homomorphisms too?

I need to prove that for an isomorphism $\phi$, the following is true: $$\phi(g)^n = 1 \iff g^n = 1.$$ We know that $$g^n = 1 \implies g\cdot g \cdots g = 1\implies \phi(g\cdot g \cdots g) = ...
0
votes
1answer
51 views

Find the Kernel of the Homomorphic $f:\mathbb{Z}_8 \rightarrow \mathbb{Z}_4$

Previously I posted a question from "A Book of Abstract Algebra" to prove that the function, $f:\mathbb{Z}_8 \rightarrow \mathbb{Z}_4$ (shown below), is a homomorphism. $f = (0 \rightarrow 0, 1 ...