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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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 $(x-k)^2$. For ...
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13 views

Combintorial 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 types of matrices .
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1answer
12 views

Ring with One sided Zero divisor

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

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

Why prime avoidance lemma allows only at most 2 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} I_j), ...
6
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Studying math all day and really young

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
20 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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26 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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19 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
29 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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24 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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1answer
31 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
25 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 ...
2
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2answers
37 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
30 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
26 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
18 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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1answer
12 views

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
26 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 ...
2
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2answers
20 views

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
32 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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33 views

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
38 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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24 views

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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23 views

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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32 views

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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35 views

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 ...
3
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1answer
45 views

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 ...
2
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1answer
44 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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21 views

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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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
37 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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17 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
60 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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35 views

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 ...
2
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22 views

Image of homomorphism with kernel $N$ isomophic to $M/N$

Let $M$ be an $R$-module and $N$ be a submodule. Let $f:M\rightarrow M'$ be a surjective homomorphism with kernel $N$. How to show that $M'$ is isomorphic to $M/N$? My thought: Define ...
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2answers
66 views

Prove that $a+ib$ is prime in $\Bbb Z[i]$, of $a^2+b^2$ is prime in $\Bbb Z$.

Prove that $a+ib$ is prime in $\Bbb Z[i]$, of $a^2+b^2$ is prime in $\Bbb Z$. My Try: We can easily show that $\Bbb Z[i]$ is a FD but how can we show that $\Bbb Z[i]$ is a UFD. Because if we can show ...
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Simple composition factor

We know that if $e$ is a primitive idempotent in a semiperfect ring then $Re/Je$ is a simple left $R$-module, where $J$ is the Jacobson radical of $R$. Now, suppose that $e$ is an idempotent in an ...
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1answer
21 views

To show module homomorphism being injective

Suppose $R$ is a field, $M$ is a right $R$ module, and $f : R_R \rightarrow M$ is a non-zero homomorphism. Show that $f$ is injective. My work: The homomorphism is uniquely determined by $f(1)$. If ...
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42 views

Show that $2$ is not prime in $\mathbb Z[\sqrt{-d}]$ for odd prime $d$

I have to show that if $d > 2$ is a prime number, then $2$ is not prime in $\mathbb Z[\sqrt{-d}]$. The case when $d$ is of the form $4k+1$ for integer $k$ is quite easy: $2\mid ...
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1answer
29 views

given polynomial has a root in $Z_p$…

To check $f_x$ is irrudicble or not in $f_p$ check wheather 0,1,2 , p-1 is a root of $f_x$...if any of this is a root then it is not irrudicible.. is this method applicable here?
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1answer
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How many possible isomorphisms do we have between G and H? [duplicate]

Let $G=(Z_4,+)$ and let $H=(U_5,*)$ where $U_5 = \{[1],[2],[3],[4] \}$ . I know that $[1]$ and $[3]$ are both generators for $G$. I also know that $[2]$ and $[3]$ are both generators for $H$. In order ...
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37 views

Should a reflection matrix of a vector have the same form as a rotation matrix?

According to the book: I know that it is not possible to write a reflection as a rotation, but from the text it seems that the matrix of the form $$ A=\left[ \begin{array}{ c c } a & ...
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2answers
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The abelianness of the quotient group of an abelian group.

I am working on an assignment for my abstract algebra class. The question states: Let $A$ be an abelian group and let $B$ be a subgroup of $A$. Prove that $A/B$ is abelian. I was under the ...
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43 views

Isomorphisms between $(\mathbb Z_4,+)$ and $(U_5,*)$

So I am asked to find all the isomorphisms between $G = (\mathbb Z_4,+)$ and $H = (U_5,*)$. I solved it as follows: we will have two isomorphism corresponding to the two generators of $U_5$. The ...
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Find up to isomorphism all the quotient groups of composition series of a group of order $30$.

I can't seem to understand what I should do here... All I did so far is proving that $G$, (such a group), is not simple. But there are many cases, I can't really tell what $n_2,n_3$ and $n_5$ are, ...