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Questions tagged [abstract-algebra]

Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, and modules, among other topics. Whenever you use this tag, please also include related tags like group-theory, ring-theory, modules, etc., in order to clarify ...

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Defining the rank for a finitely generated abelian group

So first some definitions. Let $G$ be an abelian group, a basis for $G$ is a linealry independant subset that generates $G$. We say that $G$ is finitely generated if a basis for $G$ is finite. Now ...
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Do there exist two finite groups $H$ and $K$, satisfying specific conditions?

Let’s define $\sigma(G)$ as the sum of orders of all normal subgroups of a finite group $G$. Do there exist two finite groups $H$ and $K$ such, that $\sigma(H) = |H| + |K| = \sigma(K)$, $|H|$ is even ...
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0answers
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Cutting off an infinite matrix(Making a finite matrix from an infinite matrix)

Let's say that there is an infinite matrix A. How do you make a new finite matrix B, from the matrix A? What I mean is that I want to get a finite matrix that best matches/approximates the original ...
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0answers
15 views

Torsion Coefficient in Group Theory

I have seen a calculation about torsion coefficient Determining torsion coefficients But I am now facing another similar question and not sure about is my answer correct. The question: Find the ...
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1answer
44 views

$H \leq \mathbb{Z}_q^n$ and $H \cong \mathbb{Z}_q^m$ implies that $\mathbb{Z}_q^n / H \cong \mathbb{Z}_q^{n-m}$

Given a pair of positive integers $n,q$ and a subgroup $H \leq \mathbb{Z}_q^n$ such that $H \cong \mathbb{Z}_q^m$ for a positive integer $m < n$ then show that $$ \mathbb{Z}_q^n / H \cong \...
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0answers
18 views

about group law on extended square class group

This is excerpt from W. Scharlau: Quadratic and Hermitian Forms, page 37. In this $Q(K)$ we define group law as given I don't understand what is third law saying (1,$\alpha$)+(1,$\beta$)=(0,-$\alpha$$...
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Linear Diophantine Equation Signs

How to determine the coefficient signs for the solution of a linear diophantine equation? Take $24x + 69y = 33$ for example. I know the solution is $x = 33 − 23k , y = −11 + 8k$, and I understand ...
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1answer
42 views

Cyclic notation for residue classes mod $n$

I am reading an example from Dummit and Foote's Abstract Algebra, 3rd edition. I am wondering why the authors do not use the bar notation for the element of residue classes. For example, why do they ...
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0answers
24 views

A Euclidean-like property for partially-ordered magmas

Let $\mathbf{X} = (X,+,\leq)$ be a partially ordered magma, i.e. $(X,+)$ is a magma (written additively), and $\leq$ is a partial order on $X$ satisfying: for every $x,y,z \in X$ such that $x \leq y$, ...
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0answers
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Are finite extensions of global function fields always simple? [duplicate]

A global function field is a finite extension of $\mathbb{F}_p(T)$ for some prime number $p$. A finite field extension $L/K$ is called simple if it has a primitive element, i.e. there exists an $x \in ...
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1answer
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Clarification of the proof of the Second Isomorphism Theorem

I am reading the proof of the Second Isomorphism Theorem on Dummit and Foote's Abstract Algebra. Could someone please explain how $\varphi$ is surjective? If $(ab)B$ is any element of $AB/B$, I don'...
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1answer
35 views

Proving $(A + I) / I \cong A/(A \cap I)$

If $A$ is a subring of $R$ and $I$ is and ideal of $R$, let $A+I=\{a+i:a\in A,i\in I\}$, then prove $(A + I) / I \cong A/(A \cap I)$. We need the second theorem of homomorphisms but I need to find ...
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3answers
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What is the crime of lèse-Bourbaki?

In the foreword to his textbook Algebra, Serge Lang writes (on page vi) I have frequently committed the crime of lèse-Bourbaki by repeating short arguments or definitions to make certain sections ...
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1answer
45 views

Proof Check: Two well-ordered algebraic closures of a field $F$ are isomorphic (without Zorn's Lemma)

Let $F$ be a field that can be well-ordered. This post proved that $F$ has a well-orderable algebraic closure, and I tried to prove that any two of them are isomorphic: Proof. Let $F_1$ and $F_2$ ...
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2answers
32 views

Confusion on the definition of presentation of Groups

We define a group presentation to be an ordered pair denoted by $\langle S | R \rangle$ where $S$ is an arbitrary set and $R$ is an arbitrary subset of the free group $F(S)$. We call the elements of $...
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1answer
17 views

Is the monoid ring of a Noetherian monoid Noetherian?

Let $M$ be a monoid. Suppose that $M$ is left Noetherian, i.e. that every increasing chain of left ideals in $M$ stabilizes. Then is the monoid ring $\mathbb{C}[M]$ necessarily a left Noetherian ring? ...
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1answer
21 views

Trouble with question about ideals and homomorphisms

Having trouble with this following question: Let $R$ be a ring and $I$ and $J$ ideals of $R$ such that $J\subset I$ ($J\subseteq I, J\neq I $) a) Show that the set $I/J$ is ideal of quotient ...
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1answer
63 views

Do there exist finite non-cyclic groups $H$ and $K$, satisfying the specific condition?

Let’s define $\sigma(G)$ as the sum of orders of all normal subgroups of a finite group $G$. Do there exist two finite groups $H$ and $K$ such, that $\sigma(H) = |H| + |K| = \sigma(K)$ and $H$ is non-...
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0answers
25 views

What does mean by a Topological closure $ F' \ $ of a field $ \ F \ $ with respect to a norm $ \ ||.|| \ $?

What does mean by a Topological closure $ F' \ $ of a field $ \ F \ $ with respect to a norm $ \ ||.|| \ $ ? Answer: I know the closure of a subset S of points in a topological space consists of all ...
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1answer
30 views

Gauss' Lemma prove $\mathbb{Z}[x]$ UFD

I am trying to deduce that $\mathbb{Z}[x]$ is a UFD given the fact that the product of two primitive polynomials $fg$, given $f,g\in{\mathbb{Z}[x]}$, is primitive (I have managed to prove this myself)....
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0answers
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Continuous automorphisms of $\mathbb{C}_p$

Goodmorning my queston is: Every continuous automoprhism of $\mathbb{C}_p$ is on the form $\sigma(x)=lim_{n\to \infty}\bar{\sigma}{(x_n})$ where $x =lim_{n\to \infty}x_n $ and $\bar{\sigma}\in ...
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1answer
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Show that the property holds for the group homomorphism

We have $(\mathbb{Q}^2,+)$ with the operation $(x,y)+(x',y')=(x+x',y+y')$. For $a\in \mathbb{Q}$ and $(x,y)\in \mathbb{Q}^2$ we have that $a\cdot (x,y)=(a\cdot x, a\cdot y)$. Let $f:\mathbb{Q}^2\...
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0answers
57 views

Constant sum of characters

Let $q$ be a prime power and $\omega=\exp(2\pi i/q)$. For a fixed $y\in\mathbb{Z}_q^n$, the map $$\mathbb{Z}_q^n\ni x\mapsto \omega^{x\cdot y}=\omega^{x_1y_1+\dots+x_ny_n}$$ is a character of $\...
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1answer
37 views

Witt ring of field

Why $W(C)$ is isometric to $\mathbb{Z}_2$ and $W(R)=\mathbb{Z}$ how to get this explicitly? I know we can count $\mathbb{C}$ as even dimensional space over $\mathbb{R}$ so hyperbolic space hence it is ...
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1answer
35 views

When is the group homomorphism injective or surjective?

I am looking the following two questions: 1) For which natural numbers $m,n\in \mathbb{N}$ is there is there an injective group homomorphism $(\mathbb{Z}/m\mathbb{Z}, +)\rightarrow (\mathbb{Z}/n\...
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1answer
25 views

If $A$ is absolutely flat then every primary ideal is maximal

Given the ring $A$ is commutative with an identity element. If $A$ is absolutely flat (i.e. each $A$-module is flat) then every primary ideal is maximal. This exercise 4.3 comes from the ...
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2answers
558 views

A finite group such that every element is conjugate to its square is trivial

Suppose $G$ is a finite group such that $g$ is conjugate to $g^2$ for every $g\in G$. Here's a proof that $G$ is trivial. First, observe that if $\lvert G\rvert$ is even, then $G$ contains an ...
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0answers
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Confused with $Z_n =\langle x\rangle $

It's may be a silly question, but I found myself confused here. We know $Z_n $ be a group under addition. So if $Z_n =\langle x\rangle $ Then I can't understand how every element of $Z_n $ can be ...
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1answer
17 views

A question about ideals, maximal ideals, and UFD's.

I'm working on the following problem in preparation for an exam. Let $f(x) = x^3 + 2 \in \mathbb{Z}[x]$. Let $\alpha$ be any complex number satisfying $f(x)$. Let $K = \mathbb{Q}(\alpha)$ and suppose ...
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1answer
61 views

Are the subsets subgroups?

Let $X=\{1,2,3,4\}$ and $\text{Sym}(X)$ the group of bijections $f:X\rightarrow X$ with the composition of maps as the operation. I want to check if the following subsets $U_i$ of $\text{Sym}(X)$ ...
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2answers
58 views

Dummit and Foote problem 11 in section 7.4

I am trying to solve problem 11 in Dummit and Foote section 7.4. The problem is the following: Assume $R$ is commutative. Let $I$ and $J$ be ideals of $R$ and assume $P$ is a prime ideal of $R$ that ...
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2answers
35 views

Explain why the ideal $(x,y)$ of $\mathbb Q[x,y]$ is not generated by a single element.

Explain why the ideal $(x,y)$ of $\mathbb Q[x,y]$ is not generated by a single element. I know the ideal $(x,y)$ is maximal since $\mathbb Q[x,y]/(x,y) \cong \mathbb Q$; and I know $\mathbb Q[x,y]$ ...
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1answer
15 views

lifting generators of multiplicative groups of prime power

I'm trying to understand a proof when the multiplicative group of integers modulo n is cyclic. It starts by supposing $g + p \mathbb{Z}$ is a generator in $(\frac{\mathbb{Z}}{p\mathbb{Z}})^{\times}$. ...
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4answers
43 views

Show that in the ring $R = \mathbb{Q} [x, y]$ there are ideals that require at least two generators ( the ideal $I =\{f\in R: f (0,0) = 0\}$

Show that in the ring $R = \mathbb{Q} [x, y]$ there are ideals that require at least two generators (for example, the ideal $I =\{f\in R: f (0,0) = 0\}$) What would be the generators for the example ...
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1answer
32 views

How to prove the set of integers under addition and multiplication modulo n is a commutative ring with unity 1? [on hold]

How to prove the set of integers under addition and multiplication modulo n is a commutative ring with unity 1?
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0answers
293 views

Coefficients such that linear combination lies in an ideal

Let $R$ be a ring, $I$ an ideal, and $\langle g_1, \ldots, g_m \rangle$ a finitely generated ideal. Considering the intersection $I \cap \langle g_1, \ldots, g_m \rangle$, I became interested in the ...
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2answers
54 views

Does there exist a non-trival set $W$ such that $W\times W \cong W\times W\times W$?

Does there exist a non-trival set $W$ such that $W\times W \cong W\times W\times W$ where $\times$ is the exterior product? Consider $R^2$ and $R^3$, I don't think they were isomorphism to each other, ...
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2answers
18 views

Defining multiplication of quotient ring with respect to polynomial

Let $R=\mathbb{Z}_{11}[x]/(x^2+2)$. I would like to define multiplication in $R$. To my understanding, it is required that if $[ax+b][cx+d]=[rx+s]$ we want to find $r$ and $s$ in terms of $a,b,c,d$. ...
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1answer
25 views

Vectors and points; abstract differences [duplicate]

I would ask this on stackoverflow but I wanted a more theoretical answer. I understand that vectors have magnitude and direction whereas points are just coordinates. In computing and especially in ...
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1answer
34 views

If G is a group of 2 x 2 matrices under matrix multiplication and a,b,c,d are integers modulo 2 , ab - bc is not equal to 0 it's order is 6?

EDIT - "Sorry i got the problem . Problem is right ." The above question is from Herstein algebra .I think question is wrong .Suppose i take a=1 , b=1 , c=0 , d=1 in a matrix X , it satisfies the ...
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2answers
27 views

Understanding the definition of prime element.

I have trouble understanding the definition of a prime element. The definition says that $p$ is a prime element if $p$ divides $ab$ then either $p$ divides $a$ or $p$ divides $b$. but if we consider ...
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1answer
14 views

Find number of polynomials in $P_n(F)$ where $F$ is a finite field with cardinality $m$. [duplicate]

Firstly, I should mention that the above question hits my mind when I am solving a problem in Abstract Algebra which is- Find the number of elements in $\frac{\Bbb{Z}_3[x]}{<x^3+2x+1>}$. I have ...
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1answer
39 views

Does there exist a nearly immaculate group not of the form $C_{2^n}$?

Define a nearly immaculate group as a finite group $G$, such, that the sum of the orders of all its normal subgroups is $2|G| - 1$. It is quite obvious to see, that all groups of the form $C_{2^n}$ ...
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1answer
53 views

Homomorphism from $\mathbb R^2\to \mathbb C$

Is it possible to define a surjective ring homomorphism from $\mathbb R^2$ onto $\mathbb C$? The multiplication defined on $\mathbb R^2$ is as follows: $(a,b)(c,d)=(ac,bd)$
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1answer
42 views

Continuity of trace and norm

Let $L|K$ be a finite extension of discrete valuation fields (not necessarily complete). Consider the classical maps: $$\operatorname{Tr}_{L|K}:L\to K$$ $$N_{L|K}:L^\times\to K^\times$$ Are such ...
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1answer
51 views

$\operatorname{char}(F)=0$ or a prime number

I have been taught that the characteristic of a field ($F$) is always $0$ or a prime number, and that this proves it: Assume $\operatorname{char}(F)=ab\space, a,b>1$ $\operatorname{char}(F)$ is ...
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1answer
46 views

Proving that $\operatorname{Tor}_n^R$ is a bifunctor

$\newcommand{\Tor}{\operatorname{Tor}}$ Ex10.2 pg 615: For a ring $R$ and fixed $k \ge 0$, prove that $\Tor_n^R(-,-)$ is a bifunctor. I am aware of this post. I am also not satisfied with the ...
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1answer
42 views

Non-unital ring contains a unital ring as a subring?

It is well known that every non-unital ring can be embedded into a unital ring (e.g., Dorroh's adjunction). I am curious about the converse: every unital ring can be viewed as a subring of a non-...
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0answers
17 views

Need explanation of a statement from Gelbert book of automorphic forms on Adele groups.

I was going through the book Automorphic forms on Adele groups by Stephen S. Gelbert and in the second page of first chapter I got a statement like all Fuchsian groups have a finite numbers of Γ-...
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1answer
18 views

Naturality of torsion distributing over direct sum

On pg 408, proposition 7.6, Rotman states, If $(B_k)$ is a family of left $R$-modules, then there are natural isomorphisms, for all $n \ge 0$, $$Tor_n^R(A, \bigoplus_k B_k) \cong \bigoplus_k ...