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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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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456 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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87 views

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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302 views

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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36 views

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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32 views

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 is the Difference Between an Algebra and a Field?

The Wikipedia page for $\sigma$-algebra, as well as some other resources I'm studying, say this set is called a "sigma-algebra" by some, and called a "sigma-field" by others. I'm writing a paper on ...
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33 views

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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33 views

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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42 views

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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65 views

$x^3+nx+2$ 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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Presentation of an object in an Eilenberg Moore category by generators and relations

Let $\cal C$ be a category and let $T \colon \cal C \longrightarrow C$ be a monad. An $T$-algebra $A$ is presented by generators $G \in \cal C$ and relations $R \in \cal C$ if it is the coequaliser of ...
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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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66 views

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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30 views

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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73 views

Why is the order of the subgroup 3?

I want to find the order of the subgroup $\langle ab\rangle$ of $D_3=\langle a,b\mid a^3=1,b^2=1,ba=a^2b\rangle$ According to my notes, the order of this subgroup is 3. But why is it like that? I ...
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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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About $Z_{p}[\sqrt{k}]$, when is it a field?

I give up. I'm new in the fields world, and I'm trying to give a sufficient and necessary condition for $\mathbb{Z}_{p}[\sqrt{k}]=\{a+b\sqrt{k}:a,b\in \mathbb{Z}_{p}\}$ to be a field ($p$ is a prime ...
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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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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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81 views

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 ...
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Is $\mathbb{Z}_9^*$ cyclic?

I'm reading "A course in algebra" (2003) by E. B. Vinberg to have a basic understanding on algebra. Now I got a question. Exercise 4.46. says "Prove that the group $\mathbb{Z}_n^*$ of invertible ...
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Inverse limit of groups

Can we replace topological spaces by groups in Corollary 1.1.6 from the this link? In other words, does the following theorem is true? Let $Y$ is the inverse limit of system of groups ...
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If $f(x)\in F[x]$ is solvable by radicals then its Galois group $Gal(E/F)$ is solvable.

In all the books that I've checked the use of roots of unity (as hypothesis) is very crucial to prove that if $f(x)\in F[x]$ is solvable by radicals then its Galois group $Gal(E/F)$ is solvable, but ...
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Proving a simple claim concerning order without using LaGrange's Theorem

For whatever reason, I am having trouble proving the following claim without using LaGrange's Theorem. Claim: Let $G$ be a group of order $n < \infty$. Then, $x^{n}=1$, where $1$ is the identity, ...
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Subgroups of Symmetric Group $S_4$ and Isomorphism

During my Algebra class we were given this exercise to solve at home, but I couldn't find any solution and I also did not really get the one our teacher gave us. So, the text was: Given G = S4 = ...
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When is $N\otimes_A B \to N$ an isomorphism?

Let $A, B$ be commutative (unital) rings and $f\colon A \to B$ an $A$-algebra. There then exists a canonical functor $f_*\colon \mathbf{Mod}_B \to \mathbf{Mod}_A$ such that, for every morphism of ...
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Projective and simple modules over finite dimensional algebras

I'm working through some lecture notes on the representation theory of a finite dimensional algebra $A$ (associative, unital, over an algebraically closed field $k$), and have got stuck on a ...
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For which categories do injections induce surjections in cohomology?

I'll ask a specific question first, but I believe my question might have a rather immediate abstraction, with which I'll finish. Let $H,G$ be finite affine group schemes over an algebraically closed ...
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Any simple proof of that localization of a UFD is a UFD without using Kaplansky condition?

Using Kaplansky condition, the proof of this statement is quite easy. Before knowing this condition, I tried to prove this statement by actually showing the definition of UFD holds on $S^{-1}A$ ($A$ ...
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$\beta_a(n)=(a_1*\cdots(a_n*b))\setminus_* b$ and Iterations in right divisible magmas e representability by left translations.)

Let's consider the magma $(G,*)$ with infinite elements. Now I define $\operatorname{left}(G)$ the set of all the left translations $$\operatorname{left}(G):\{L_a:a \in G ,L_a(b)=a*b\}$$ And ...
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70 views

exact sequence and modules proposition.

I have problems to prove the following proposition: Let's consider $$0 \rightarrow L \stackrel{\alpha}{\rightarrow} M \stackrel{\beta}{\rightarrow} N \rightarrow 0$$ an exact sequence of modules and ...
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Frobenius on projective variety is not an isomorphism?

This is exercise 1.8 from Arithmetic of Elliptic Curves (Silverman). Part 3 confuses me because isn't $\phi$ the identity, thus an isomorphism? Let $\mathbb{F}_q$ be a finite field with $q$ ...
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Elements of a Dedekind domain can be chosen to have valuation $1$ with respect to one prime, $0$ everywhere else

I noticed this is true for $\mathbb{Z}$, but I was wondering whether it was true in general. Let $R$ be a Dedekind domain and $P_1, ... , P_s$ maximal ideals. The localized ring $R_{P_i}$ is a ...
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primes splitting completely in cyclic extensions

Let $K$ be a quadratic number field. It is a well known result that a prime $p$ splits completely in $K$ if and only if $\left(\frac{d_K}{p}\right)=1$. What about cubic extensions? Can we find ...
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50 views

If $p>2$ is a prime number, Prove that $U(p^k) $ is cyclic

(i) If $p>2$ is a prime number, Prove that $U(p^k) $ is cyclic (ii) Prove that $U(p^n) \thickapprox Z_{(p^n -p^{n-1}) }$ Solution : If I prove that if $U(p^k) $ is cyclic, then, we know that ...