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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Algebraic values of the sine function

First question: For which angles $x$ is $\sin(x)$ a real number that can be expressed using only integers, addition, subtraction, multiplication, division and the extraction of $n$th roots? (With ...
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Clarification about the definition of free module

I am reading this notes. Let $R$ be a commutative ring with $1$. Let $S$ be a set. A free $R$-module $M$ on generators $S$ is an $R$-module $M$ and a set map $i:S\rightarrow M$ such that, for ...
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1answer
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Mazur's theorem-abelian group of rational points of an elliptic curve

From Mazur's theorem we have the following: If $E |_{\mathbb{Q}}: y^2=x^3+ax+b, a, b \in \mathbb{Z}$ an elliptic curve, then $$E(\mathbb{Q})_{\text{torsion}} \cong \mathbb{Z}/n\mathbb{Z}, \text{ for ...
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1answer
32 views

$\sqrt{I}+\sqrt{J}=R$ implies $I+J=R$

Let $R$ be a commutative ring with unity and $I,J$ ideals of $R$. Suppose that $$ \sqrt{I}+\sqrt{J}=R $$ I want to show that this implies $I+J=R$. Take $r\in R$, then I can write $$ r=a+b, $$ for ...
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33 views

Extension field of $x^3 - 2$ in $\mathbb Q$

I know this has been asked previous on Stackexchange, but I just want to have something clarified: I am supposed to find an extension K of Q having all roots of $x^3 - 2$ such that $[K:\mathbb Q] = ...
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3answers
35 views

Computing the galois group of $x^3+3x+1$

I want to calculate the galois group of the polynomial $x^3+3x+1$ over $\mathbb Q$. And I am struggling in finding the roots of the polynomial. I only need a tip to start with. Not the full solution ...
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1answer
28 views

find about center of G s.t H is normal subgroup of order 2

Let G be a finite group and H be normal subgroup of order 2. Then order of center of G is 0 1 Even integer $\ge $2 Odd integer $\ge $3 I tried this problem by taking G as $S_3$ and H as $ A_3$, ...
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2answers
32 views

Kernel and Image of a group homomorphism

let $G$ be a multiplicative group of non-zero complex analysis.consider the group homomorphism $\phi:G\rightarrow G$ defined by $\phi(z)=z^4$. 1.Identify kernel of $\phi=H$. 2.Identify $G/H$ My ...
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1answer
16 views

fundamental group of tow circles joined by an arc [on hold]

What is the fundamental group of two circles joined by an arc. In other word, let $S_1$ and $S_2$ be two standard circles. Let $p_1$ and $p_2$ be two points in $S_1$ and $S_2$ respectively. Join $p_1$ ...
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1answer
26 views

How can I prove this about the tangent line formula??

The equation of a tangent line to $f(x)$ at $x = t$ is $y = f'(t)(x - t) + f(t)$. Recently, I heard that it is also determined by the remainder of polynomial division of $f(x)$ by $(x-t)^2$. For ...
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32 views

Combinatorial approach to calculate determinant

Suppose you have set of $n*n$ matrices with entries from the set $\{1,-1\}$. Then what can be the maximum determinant which you can obtain from such type of matrices.
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1answer
23 views

Ring with One sided Zero divisor [on hold]

Does there exist ring whose all elements are left zero divisor but only one element is right zero divisor
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25 views

Why prime avoidance lemma allows only at most 2 prime ideals?

Why prime avoidance lemma allows only at most 2 non-prime ideals? The following is the last part of the proof taken from wikipedia: For the case n > 2, choose $z_i \in E \cap (I_i - \cup_{j \ne i} ...
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5answers
467 views

Studying math all day and really young [on hold]

I am very young and want to learn algebra and calculus for fun. What should I keep in mind when I start learning? I am going to try the textbooks I have borrowed out: Dummit and Foote and Spivak's ...
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2answers
27 views

Congruence definitions equivalence

We say that $x$ is congruent to $y$ modulo $z$ when $$x\equiv y\pmod z \iff x \pmod z = y \pmod z$$ Another definition is $$\quad x \equiv y \pmod z \iff \exists k \in \mathbb{Z}: x - y = k z$$ Why ...
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31 views

Two subgroups $H$ and $K$ of permutation group($G$).such that $H$ is normal in $K$ and $K$ is normal in $G$ but $H$ is not normal in $G$

Is there exist any subgroups $H$ and $K$ of permutation group($G$).such that $H$ is normal in $K$ and $K$ is normal in $G$ but $H$ is not normal in $G$?
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21 views

e Online source for alternative proofs

I'm looking for some alternative proofs for various theorems. My goal is to compile a list of various proofs each relating to a specific theorem (such as the triangle inequality, Fermat's Little ...
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2answers
38 views

Cremona group of $\mathbb{P}^n$

I know that the complex conjugation $\tau: \mathbb{P}^n \mapsto \mathbb{P}^n$ that sends any point $x$ to the point with complex conjugate coordinates $\tau(x)$ is a homeomorphism. In order to show ...
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0answers
12 views

Nilpotent Jacobson radical of $End(M)$

If $M$ is a Noetherian injective left $R$-module, is it true that the Jacobson radical $J=J(End(M))$ of the endomorphism ring of $M$ is nilpotent? I know that if a left $R$-module is Artinian or ...
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11 views

A free direct sum of a projective module

I want to prove that a left module $_RP$ is a projective generator if and only if a direct sum of (copies of) $P$ is free. My try is, first, to observe that $P$ is a generator if and only if for some ...
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1answer
32 views

Let R be a ring with unity $1_R$ and let S (with unity $1_S$) be a subring of R. Prove that either $1_S=1_R$ or $1_S$ is a zero divisor of R.

Let R be a ring with unity $1_R$ and let S (with unity $1_S$) be a subring of R. Prove that either $1_S=1_R$ or $1_S$ is a zero divisor of R. My attempt: Let $a \in S$. Then $1_S*a = a$ and this a ...
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2answers
34 views

Change of Basis for $2\times2$ matrix

Suppose I have the matrix basis $\begin{bmatrix}1&0\\0&0\\\end{bmatrix}$ , $\begin{bmatrix}0&1\\0&0\\\end{bmatrix}$ , $\begin{bmatrix}0&0\\1&0\\\end{bmatrix}$, ...
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1answer
26 views

Vector spaces and multiplicative inverse?

Do vector spaces have multiplicative inverses? They seem to be monoids under $+,\times$, so monoids $(\Bbb F, +)$ and $(\Bbb F, \times)$ where $\Bbb F=\Bbb R \,or\, \Bbb C$ And it is even a group ...
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2answers
46 views

Example of a Non-Commutative Division Ring With Finite Characteristics

Wedderburn's Little Theorem says that every finite Division Ring is commutative. What is about an infinite Division Ring with prime characteristics? Is this also a Field?
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3answers
31 views

Prove that the ideal generated by $x^3 + x + 1$ is not maximal in $\mathbb Z_3[x]$

This is part of a larger homework problem. I am trying to prove that a quotient ring is not a field by showing that $\langle x^3+x+1\rangle$ is not maximal in the ring of polynomials in the integers ...
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2answers
30 views

Ideals of formal power series ring

I need help understanding the following solution for the given problem. The problem is as follows: Given a field $F$, the set of all formal power series $p(t)=a_0+a_1 t+a_2 t^2 + \ldots$ with $a_i ...
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0answers
20 views

Prove: given a ring R with left identity $e_l$ and the right identity $e_r$, then $e_l = e_r$. Another way to prove?

Suppose a ring R has the left identity ($e_l$) and the right identity ($e_r$). Then $e_l = e_l*e_r = e_r$. I was wondering if there's another way to do it. Thank you.
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abstract algebra question about integral domain

Prove that any subring of a field which contains the identity is an integral domain. Do i need to show that $ab = 0$ where $a = 0$ or $b = 0$. Or in general What do you need to show in order ...
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1answer
27 views

Why is it not a sufficient condition to conclude that a is a unity based only on the information that $xa = x$ for all $x$ in $R$?

We have a ring $R$ as follows: Why is it not enough to conclude that $a$ is a unity if $xa = x$ for all $x$ in $R$? Is it because it is by definition that the unity satisfies $ax = xa = x$ for all ...
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2answers
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How to show the isomorphism

I'm not sure how to show this $$\mathbb{Z}_3[X]/(x^3 -x +1)\cong\mathbb{Z}_3[X]/(x^3 -x^2+x +1).$$ Any help or hint would be helpful.
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1answer
35 views

Classifying groups of order 60

I want to solve the following problem from Dummit & Foote's Abstract Algebra text (p. 185, Exercise 14): This exercise classifies the groups of order $60$ (there are thirteen isomorphism ...
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Coproduct of groups

Can anyone explain why the coproduct of groups are the free product? For finite groups, the products are direct products which are also finite. But the free groups are infinite? So the coproduct is ...
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3answers
39 views

Prove or disprove: For every integer a, if a is not congruent to 0 (mod 3), the a^2 is congruent to 1 (mod 3)

Prove or disprove: For every integer a, if a is not congruent to 0 (mod 3), the a^2 is congruent to 1 (mod 3) SO this is for abstract algebra and I am really struggling with this. Here are some of ...
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Objects whose limiting behaviour resembles a group

Is there a name for a structure that isn't a group, but that begins to behave like a group the more operations are performed? I'm trying to take the idea of an attractor from dynamical systems and ...
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Question about normal subgroups and cosets

I have seen two definitions of a normal subgroup. The first one: A subgroup $N$ is normal to a group $G$ if $xN = Nx$, $\forall x \in G$ The second one: $N$ is normal to a group $G$ if $N ...
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Finding a Basis for polynomial subspace

This is problem 14 in Herstein's Topics in Algebra. I'm having trouble with the problem (working through the text independently). For $F$ a field, define $V_n=\{p(x)\in F(x) : \deg p(x)<n, n\in ...
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1answer
20 views

Commutative matrix question

I was doing my HW, and I am confused with one thing. To show that a matrix is commutative, do we need to show both $x+y = y+x$ and $xy=yx$? Or just by showing $xy=yx$ would suffice?
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Their product is a cubic of a rational number $x$ minus $x$

It is given the integer $6$. Analyze it into two parts such that their product is a cubic of a rational number $x$ minus $x$. $$$$ Let $y$ be the one factor. The other one is $6-y$. We have ...
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1answer
16 views

Is there a general way to find what $Aut(C_n)$ Is Isomorphic to?

I'm asked to describe $Aut(C_{21}),Aut(C_{24})...$ as a product of cyclic groups - but I'm wondering is there a general way to do this?
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Automorphisms of Abelian groups

Let $A$ be a free Abelian group and $N$ a characteristic subgroup of $A$ such that $A/N$ is finite. I also know that $Aut(A/N)$ and $Aut(N)$ are both finite. I have to prove that $Aut(A)$ is finite. ...
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Why we throw away the units in the definition of irreducible elements?

In the book "Abstract Algebra" by Dummit, the definition of irreducible element in an integral domain $R$ goes like this. Suppose $r\in R$ is nonzero and is not a unit. Then $r$ is called irreducible ...
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Abelian torsion group of rational points of an elliptic curve

I want to find the abelian group of rational points $E(\mathbb{Q})_{\text{torsion}}$ of the elliptic curve $y^2=x^3+8$. $$E(\mathbb{Q})_{\text{torsion}}=\{P \in E(\mathbb{Q}) | P \text{ of finite ...
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1answer
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Can every group be extended to ring with idenity [duplicate]

Can every abelian group converted into ring(by defining multiplication operation) with identity with same order. We can convert every group G into ring by defining a.b = 0 for all a and b in G. But ...
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1answer
45 views

Every finite Set as non-abelian Group

For what values of n, we can find a non abelian group. The facts I have proved till now: 1. For n prime there exist only one group upto isomorphism which is cyclic hence abelian 2. For n = 4, there ...
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An integer $m$ is a prime element in $\Bbb Z[i] $ if $m$ is a prime number of the form $4n+3$ [duplicate]

An integer $m$ is a prime element in $\Bbb Z[i] $ if $m$ is a prime number of the form $4n+3$. I am stuck with the proof....please help!
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1answer
113 views

Is $\text{Hom}(\prod_p \Bbb Z/p\Bbb Z, \Bbb Q) = 0$ possible without choice?

That divisible abelian groups are precisely the injective groups is equivalent to choice; indeed, there are some models of ZF with no injective groups at all. Now, given that $\Bbb Q$ is injective, ...
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1answer
41 views

Abelian group of rational points of an elliptic curve

I want to find the abelian group of rational points $E(\mathbb{Q})_{\text{torsion}}$ of the elliptic curve $y^2=x^3-2$. $$E(\mathbb{Q})_{\text{torsion}}=\{P \in E(\mathbb{Q}) | P \text{ of finite ...
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0answers
19 views

Module product and coproduct

Completely lost when reading this: "Defintion: If the index set $T$ is finite, the coproduct and product are the same module. If $T$ is infinite, the coproduct or sum $$\coprod_{t\in ...
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2answers
61 views

How to show that $3\mathbb{Z}/15\mathbb{Z} \cong \mathbb{Z}/5\mathbb{Z}$?

How to show that $3\mathbb{Z}/15\mathbb{Z} \cong \mathbb{Z}/5\mathbb{Z}$ as $\mathbb{Z}$-module over $\mathbb{Z}$? My proof: Define surjective function $f:3\mathbb{Z}/15\mathbb{Z} \rightarrow ...
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Inverse property for groups Proof

I was wondering if (1) this proof is correct, and (2) if other proofs exist for the following: Prove that $(a_1a_2...a_n)^{-1}=a_n^{-1}a_{n-1}^{-1}...a_1^{-1}$ where $a_i \in $ a Group $G$ Proof by ...