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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On the Nakayama functor

Let $A$ be a finite dimensional $k$-algebra with 1. Denote by $_AP$ the category of projective left $A$-modules finite dimensional. And with $_AI$ the category of injective left $A$-modules finite ...
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Profinite group power map

I am working through some exercises in Neukirch's Algebraic number theory and i need some hints for exercise 1 pg 274, it goes as follows: Let $G$ be a profinite group, show that we can extend the ...
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Are there any infinite dimensional subalgebras of the Witt algebra?

The Lie bracket of elements of the Witt algebra is given by: $[L_m,L_n]=(m-n)L_{m+n}$ Are there any infinite dimensional subalgebras of the Witt algebra that are not isomorphic to the Witt algebra ...
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Split short exact sequences and the associated graded algebra

Let $G$ be a group and $K$ be a field. Let $K(G)$ be the group ring and $I$ be its augmentation ideal. You are given that there exists a section of the projection $\pi_s:I^s\to I^s/I^{s+1}$ for all ...
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Ext in Dedekind domains

I know and can prove that $\operatorname{Ext}_Z^1(\mathbb{Z}/p\mathbb{Z},A) \simeq A/pA$. Does similar formula work for more general rings, such as Dedekind domains and their ideals, i.e. ...
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203 views

Zero divisors in modules?

Let $R$ be a ring. I find myself considering $M=R^n$, as an $R$-module. If $a$ is not a zero divisor in $R$, it holds that $\forall x \in M: ax=0 \Rightarrow a=0 \vee x=0$ . For what kinds of ...
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Correspondence between summands of a module and primitive idempotents of the endomorphism ring of the module

I am reading this paper and have the following questions. Let $A$ be a finite-dimensional algebra over a fixed field $k$. Let the finitely-generated $A$-module $M$ be a generator–cogenerator for ...
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Curve - non singular curve and its genus

Help me please with this problem:$ X \subset \mathbb{P}^{2}$ defined as $x^{3}y+y^{3}z+z^{3}x=0$ 1.Prove X - non singular curve and find its genus. 2.Prove X - maximal curve over $F_{8}$ field, and ...
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Matrix group as subgroup of $GL(n,\mathbb{R})$ or $GL(n,\mathbb{C}$)?

As far as I understand, one has (at least?) two choices to introduce infinite matrix groups: Either, one can say they are all subgroups of the general linear group over the complex numbers numbers ...
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Is there a formula for $|\langle H\cup K\rangle|$, if $H$ and $K$ are nonempty subsets of a finite group $G$?

I know that if $H$ and $K$ are subgroups of a finite group $G$, then $|HK|=\frac{|H||K|}{|H\cap K|}$. Is there a formula for $|\langle H\cup K\rangle|$, if $H$ and $K$ are nonempty subsets of a ...
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Homogeneous polynomial in $n$ variables of degree greater than $n(n-1)$ is in the ideal generated by the elementary symmetric polynomials

Let $f(x_1,...,x_n) \in \mathbb{Z}[x_1,...,x_n]$ be homogeneous of degree $d>n(n-1)$, i.e. $f$ is the sum of monomials of degree $d$. I am looking for a hint to prove that $f$ is in the ideal ...
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475 views

What does a Cayley table tell?

Other than showing if a group is abelian and the order of the group, what else does a Cayley table indicate?
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How do I find the orbits given the ring of invariants?

Suppose I have a group $G$ acting linearly on a vector space $V$ and I know the ring of invariants $K[V]^{G}$ (i.e., I know the subring of $K[V]$ which is fixed pointwise under the induced action). My ...
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The cohomology of finite $G$-modules

This is to some extent a continuation of an earlier question of mine. Now that I'm all cleared up on what it means for a finite group to have periodic cohomology, I have another question; first I will ...
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Is there an integral domain with a lot of residue fields of the same characteristic?

Is there a commutative integral domain $R$ in which: every nonzero prime ideal $Q$ is maximal, and for every prime power $q\equiv 3 \bmod 8$, there is a maximal ideal $Q$ of $R$ such ...
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A finite length module is the direct sum of the image and kernel of a projection-like endomorphism

I have a question about abelian groups (or rather $X$-groups): Suppose $A$ is an abelian group (or rather an $X$-group) which is artinian and noetherian. $f$ is an endomorphism (or rather an ...
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Prove that an equation has no elementary solution

There are methods proving that a polynomial isn't solvable in radical extensions (see Abel–Ruffini theorem) or proving that an integral or a differential equation has no solutions expressible through ...
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How to solve this equation in radicals?

How to solve the equation $x^6-2\varphi^5x^5+2\varphi x+\varphi^6=0$ in radicals? where $\varphi$ is the golden ratio.
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How do I show that these two presentations are isomorphic?

I'm taking algebraic topology this year and my professor assigned a class an exercise, and he told that the exercise will be on a coming exam. The exercise is: Show that $(x,y|xyx=yxy)\cong ...
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Central extension: $Hom(Q,Z) \cong$ automorphisms of $G$ acting trivially on subgroup and quotient?

If we have a short exact sequence $1 \rightarrow C \rightarrow G \rightarrow Q \rightarrow 1$ where $C$ is central in $G$ and $Q \cong G/C$, how can I find an isomorphism between $Hom(Q,C)$ (which is ...
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When do Leibniz-like rules lead to unique linear operators

Background Usually one defines differentiation in terms of limits, and then shows that differentiation satisfies the Leibniz (product) rule, $$\frac{d}{dx}(f \cdot g) = f\frac{dg}{dx} + g ...
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Is the identification between symmetric tensors and homogeneous polynomials useful?

The general question: Given an $n$-dimensional vector space $V$ over a field $k$, there exists an identification $$\mathrm{Sym}^d(V) \sim k[x_1, \dots, x_n]_d$$ between the space of symmetric order ...
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Restricting a representation to a subgroup

This little factoid from algebra quals stumped me: Let $G$ be a finite group and $H \triangleleft G$ an index $2$ subgroup. If we take an irreducible complex representation $V$ of $G$ and restrict it ...
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Abelian group action exercise

Let $X$ be a set with $n$ elements and let $G$ be an abelian group acting on $X$ such that: $$(i) \space gx=x \space \forall x \implies g=1,$$ $$(ii) \space \forall x,y \in X, \exists g: gx=y.$$ Show ...
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find these linear functionals

I'm trying to solve this question: My attempt of solution: If $x\in E$, see $x$ in the first $m$ coordinates of $\mathbb R^n$ (can we do this?). I know how to find linear functionals such that ...
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Why are centers, centralizers and normalizers called that way?

I know what they are, but where do the names come from?
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Showing that $\mathbb{H}^{*}$ maps onto $\mathrm{Aut}(\mathbb{H})$

To show that $\mathbb{H}$ maps onto $\mathrm{Aut}(\mathbb{H})$, where $\mathbb{H}$ is the Hamilton quaternions, I thought it'd be pertinent to show first that the subgroup of inner automorphisms of ...
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If a finite group G is nilpotent then G/F is abelian, where F is the Frattini subgroup of G

Let $G$ be a finite nilpotent group. Consider $F$ the Frattini subgroup of $G$, that is, intersection of all maximal subgroup of $G$. Prove that $G/F$ is abelian. What I am trying that G/F is ...
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Matrix representation associated to a permutation representation

I have just begun studying group representation theory. I don't understand how I can find the matrix representation associated to a permutation representation. Should I identify each permutation ...
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Resultants of two polynomials over a ring

Let $k$ be a field $f,g\in k[x,y]$ be two polynomials. The resultant $R\in k[x]$ is a polynomial function of the coefficients of $f$ and $g$, such that $f$ and $g$ gave a common zero (in an extension) ...
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Non-UFD that there exists set $X$ of any cardinality and $a$ that $xy$ is not divisible by $a^2$ for any $x,y \in X$ and $x^2$ is divisible by $a^2$

Let's say we want to construct a non-UFD that is a commutative ring that satisfies: There exists $a$ such that there exists a set $X$ of elements of finite cardinality $k$ such that for any $x,y \in ...
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Ring of rational power series

Let $A$ be any commutative ring with 1. A power series $f\in A[[t]]$ is called rational if we can find a $g\in A[t]$ such that $fg\in A[t]$. It is clear that the set of rational power series forms a ...
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What are the algebraic and topological properties of a tree (graph theory), and what are any possible connections?

I'm a non-mathematician working with trees (mostly rooted and oriented trees). Typically, I understand them as join-semilattices, so they have an implicit algebraic structure: they form a subgroup ...
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Detect cyclotomic polynomials

I was reading this question: When does a polynomial divide $x^k - 1$ for some $k$? I followed the procedure given by Bill Dubuque in his answer (the "Graeffe" method) for the polynomial $f(x) = x+1$. ...
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How can I compute |R/I| where $R\simeq\mathbb Z^n$ and $I\unlhd R$?

A number field $K$ is a field $\mathbb Q\le K\le\mathbb C$ of finite degree over $\mathbb Q$, say $n$. Call $\mathbb A$ the ring of algebraic integers of $\mathbb C$; an algebraic integer is an ...
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$x^3+nx+2$ is irreducible over ${\mathbb Z}$ for $n\neq 1,\ -3\ -5$

I want to show that $$x^3+nx +2 $$ is irreducible over ${\bf Z}$ for $n\neq 1,\ -3\ -5$ By Eisenstein, if $n$ is even then it is irreducible. How can we solve odd case ? [add] Note that ...
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“Type” in specifying a ring/ field

In the definition of fields on wikipedia, it says: A field is therefore an algebraic structure $\langle \Bbb F, +, \cdot , −, ^{−1}, 0, 1\rangle$; of type $\langle 2, 2, 1, 1, 0, 0\rangle$, ...
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Tensor product of $A_n$ modules/ localisation at ring of differentials

I'm working through Coutinho's "A Primer of Algebraic D-Modules" and I've gotten stuck on the following question: Let $p \in K[x_1, \ldots ,x_n]$ be non-zero, and let $A_n$ be the Weyl Algebra. Show ...
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zeros of a simple polynomial

For $x, y \in GF[2^n]$, consider the two-parameter polynomial $P(x,y) = x \cdot y + f(x) + g(y)$, where $f$ and $g$ are arbitrary polynomials on $GF[2^n]$. Can we say anything about the number of ...
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Is the completion of a Dedekind domain a PID?

Ths is a basic question on Dedekind domains. Let $R$ be a Dedekind domain, $P$ a non-zero prime ideal of $R$. I know that the localization $R_P$ is a PID, but is it true that the completion ...
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Determining if a given equation is solvable given a set of ultra-radicals

So suppose someone is armed with the tools of standard arithmetic, exponents (and of course that comes along with roots) AS WELL AS a set of inverses for some polynomials which are not solvable using ...
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Can all of them be different?

Edit: Cross-posted to MathOverflow here (and resolved). Let $G=\{g_1,g_2,...,g_n\}$ be a group with $e=g_1$ and $n$ is odd, Set $$a_1=g_1$$ $$a_2=g_1g_2$$ $$a_3=g_1g_2g_3$$ $$a_n=g_1g_2...g_n$$ I ...
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Simple question on Sylow subgroups.

Given a finite group $G$, $|G|=p^am$, $P,Q\in\operatorname{Syl}_p(G)$, then we know that $Q=P^x$ for some $x\in G$. Thus every element of $Q$ is the image of an element of $P$ under the isomorphism ...
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What's $[L:K(\alpha)]$ here?

Let $L=\mathbb{Q}(T_1,T_2,T_3,T_4)$ be a purely transcendental extension of $\mathbb{Q}$, and percieve $S_4$ on $\{T_1,T_2,T_3,T_4\}$ as subgroup of Aut$(L)$. Write $K=L^{S_4}$. Calculate ...
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Tensor product of the spaces of quaternions and complex numbers

Let $ \mathbb{H} $ be the ring of quaternions and make the vector space $A = \mathbb{H} \otimes \mathbb{C}$ into a ring by defining $$(a \otimes w)(b \otimes z) = (ab \otimes wz) $$ for $a,b ...
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semidirect product of semigroups

Let $S,T$ be two semigroups. A function $f:S\to T$ is called multiplicative if for any $x,y\in S$ we have $f(xy)=f(x)f(y)$. If $T=S$ then $f:S\to S$ is called automorphism on $S$. A function ...
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Artin chapter 2 exercises

I've written some solutions to Artin's exercises. Can someone please verify them? Assume the equation $xyz = 1$ holds for a group $G$. Does it follow that $yzx=1$? That $yxz=1$? We first show ...
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invariants of group action by algebra automorphism

I am trying to prove the following statement, but I'm having a lot of trouble with it: Let $k$ be an infinite field. Let $A$ be a commutative $k$-algebra. Let $G$, a group, act on $A$ by algebra ...
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Counting the number of orbits for the group action

I have have to find the number of orbits for the group action given by the set of all real $2\times 2$ invertible matrices acting on $\mathbb{R^2}$ by matrix multiplication. I know that if a group ...
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How can we solve this question without brute force

If $A\in GL_n(R)$, where $R$ is a commutative ring with identity, I would like to prove $$ M=\begin{pmatrix} A & 0 \\ 0 & A^{-1} \\ \end{pmatrix}\in ...