Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, among other topics.

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Determine the group of Automorphisms of $\mathbb{Q}(\zeta_8)$, where $\zeta_8=\exp(2\pi i/8)$

I've been tasked with the following: Let $\zeta_8=\exp(2\pi i/8)$. Compute the group of field automorphisms of $\mathbb{Q}(\zeta_8)$. This seems easy enough, I assume that the problem reduces to ...
1
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1answer
23 views

Finite dimensional division algebra over C

Another abstract algebra question from my university days that has me stumped at where to start! I know what a division ring is and I think I understand what a division algebra over $\mathbb C$ is. ...
0
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0answers
14 views

Connection between $S_{e_{n}}$ and the Clifford semigroup $S$

Can someone help me to get an answer to this question: Let $S$ be a Clifford semigroup and $S'$ sub-semigroup of $S$ and if $S'_{e_{n}}$ is normal in $S_{e_{n}}$, what can we say about the $S'$( is ...
0
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2answers
23 views

How can we define that function?

I want to show that $M=\{\tau \in S_4\mid \tau (4)=4\}$ is isomorphic to $S_3$. To do that we have to consider a function $f(x)$ that gives the isomorphism of $M$ with $S_3$, i.e., we have to ...
2
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3answers
40 views

Understanding Maschke's Theorem

https://en.wikipedia.org/wiki/Maschke%27s_theorem#Proof I'm trying to understand the need for the condition 'K's characteristic does not divide the order of G' in the statement of the theorem. Where ...
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9answers
457 views

A good book for beginning Group theory

I am new to the field of Abstract Algebra and so far it's looking to me quite tough. So far I have encountered the following books in group theory - Contemporary abstract algebra by Joseph Gallian and ...
2
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1answer
23 views

Let $G$ be a $2$-group and suppose the centralizer of some element of order two has order at most four, then $G$ has maximal class

Let $G$ be a $2$-group of order $|G| \ge 4$ and $H \le G$ be a subgroup of order $2$, i.e. $|H| = 2$. Suppose we have $|C_G(H)| \le 4$. Then $G$ has maximal class. Do you know a proof of this fact? I ...
2
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1answer
29 views

In general, when does a ring have a division algorithm?

I'm working through Herstein's "Abstract Algebra" text, and am currently working through section 5. Theorem 4.5.5 introduces the division algorithm for polynomial rings over fields, which states: ...
2
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1answer
35 views

Convincing normal subgroup proof?

I wrote the following proof on an exam, I was wondering if it makes sense. Question: let $H$ be a subgroup of $G$, written in multiplicitive notation. Prove that if the coset multiplication defined ...
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1answer
19 views

Understanding Quotienting by Relations vs Quotienting by Generators

I understand the idea of a quotient algebra $A / I$ where $A$ is a $K$-algebra and $I$ is a two-sided ideal, i.e. I understand the projection map as an algebra morphism. However, I'm unsure about how ...
2
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1answer
27 views

Showing that $P(x)=x^{p-1}-1+pQ(x)$.

This comes from a problem from Imo math notes on algebraic extensions. One needs to show that $P(x)=(x+1)(x+2) \dots (x+p-1) = P(x)=x^{p-1}-1+pQ(x)$, where $p$ is prime and $Q(x)$ a polynomial with ...
3
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1answer
33 views

Are all countable torsion-free abelian groups without elements of infinite height free?

The height of an element $g$ in an abelian group $G$ is the largest $n\in \mathbb{N}$ such that there exist an element $h\in G$ such that $n*h=g$. If $g$ has no such largest integer than $g$ is of ...
0
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1answer
28 views

Software for generators of different structures such $\mathbb Z_5$ and such as $\mathbb Z_{10}/\{0\}$? [on hold]

I need generators of different things such as $\mathbb Z_5$ and $\mathbb Z_{15}/\{0\}$ to learn abstract algebra. Please notice that I currently do too many mistakes in calculating generators. For ...
8
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1answer
571 views

Semidirect product uniqueness argument for classifying groups of small order

I'm having trouble understanding the following method for determining the number of semidirect products between two groups in simple cases that arise when trying to classify certain groups of small ...
7
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2answers
176 views

Maximal Ideals in Ring of Continuous Functions

Dummit and Foote, 7.4.33(a): Let $R$ be the ring of all continuous functions $[0,1] \to \mathbb{R}$ and let $M_c$ be the kernel of evaluation at $c \in [0,1]$, i.e. all $f$ such that $f(c) = 0$. ...
2
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1answer
47 views

The root system of $sl(3,\mathbb C)$

I want to determine the root-system of the lie algebra $sl(3,\mathbb C)$. Does someone know a good (and complete) reference for this problem? I know that the root-system is $A_2$ but I want to see a ...
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1answer
38 views

Criterion for isomorphism of two groups given by generators and relations

When are two presentations of groups are isomorphic? In this post it is said: [...] find a set of generators of the first group that satisfies the relations of the second group [...] But I doubt ...
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1answer
20 views

Constructing a quotient ring of multivatiate polynomial ring in GAP

I need to construct the following ring in GAP: $$F_2(u_1,u_2) / \langle u_1^2=u_2^2=0,u_1u_2=u_2u_1 \rangle $$. This is what I tried and it didn't work: ...
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4answers
51 views

Even-odd multiplicative algebraic structure with idempotency? [on hold]

What is the algebraic structure for the multiplications of even elements and odd elements? Please notice that $o*o=o$, $e*e=e$ (idempotency) and $o\not = e$. 1st structure is such that even times ...
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1answer
85 views

Is Multiplication A System?

My textbook says: Clock arithmetic and modular systems were built upon ordinary numbers and involved familiar operations such as addition, subtraction, multiplication, and division. So ...
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1answer
55 views

If the tensor product of algebras $A \otimes B$ is unital, both $A$ and $B$ must be unital

It is clear that if $A$ and $B$ are unital algebras (over a field), then the tensor product $A \otimes B$ is also unital, with the unit being $1_A \otimes 1_B$. I came across an exercise that ...
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0answers
11 views

Sufficient and necessary conditions for representation of a ordered structure with a binary operation.

Given a structure $\mathcal{A} = (A, \succsim, \sqcup)$, where $A$ is a non-empty set, $\succsim$ is a weak order, $\sqcup$ is a binary operation on $A$, let $\mu$ be an order-preserving mapping from ...
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1answer
43 views

Finding a vector space over $F$ of dimension $m$ and $n$

The question is below: Let $V$ and $W$ be vector spaces over $F$ of dimensions $m$ and $n$, respectively. Find a basis for $L(V,W)$. This is what I have: Let$ (v_1, v_2, \ldots, v_m)$ be a basis of ...
2
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1answer
37 views

Constructing a quotient ring in GAP using structure constants [on hold]

I need to construct the following ring in GAP: $$Z_4(u) / \langle u^2-2u=0 \rangle. $$ This is what I tried and it didn't work: ...
3
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1answer
65 views

Collective name for algebraic structures

I am doing a thesis about various algebraic structures, primarely about groups, rings and modules (with maybe hint of algebras). However always having type out ALL of them constantly gets very tedious ...
3
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2answers
15 views

$il+M=1+M \implies il =1$ or $il=1+m, m\in M$, hence $I=R$

If we have $R/M$ is a field and $M,I$ are ideals of $R$ such that $M\subseteq I \subseteq R$. If we take $i\in I, i\not\in M$ we have $i+M \ne 0+M$. Since $R/M$ is a field, we have that $i+M$ is ...
4
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1answer
33 views

How do I use homomorphism theorem to show the assertion?

Show that $\mathbb Z[x]/\langle x^2-3,2x+4 \rangle$ is isomorphic to $\mathbb Z_2[\sqrt 3]$. I tried to use first homomorphism theorem, but not able to get that how should I approach.
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0answers
40 views

Quick way to classify groups of small order

This is a past Qualifying exam on Algerbra : I'm curious if there is a quick way to solve Prob 1. To me, directly classifying groups takes too much time and I think I could not handle this in time ...
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2answers
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Example of inverse semigroup with at least two idempotent elements

We say that the semigroup $S$ is inverse semigroup if for any $s\in S$ there is a unique $t\in S$ such that $sts=s,\ tst=t$. Suppose that $E(S)=\{e:\ e\in S,\ e^2=e\}$ and define $$s\sim ...
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0answers
19 views

How do I show this assertion? [duplicate]

Show that the ideal generated by $x^2-y$ is a prime ideal in $C[x,y]$. It would be sufficient if we show that $C[x,y]/<x^2-y>$ is an Integral Domain. Or is there any other way of showing the ...
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1answer
25 views

Matrix representation of Lie Algebra $B_2$

I'm writing some practical examples where to calculate the Killing form, the Cartan Matrix, Dynkin diagrams etc. Does anybody have on or two nice matrix representations of the $B_2$ Algebra? It would ...
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0answers
25 views

How to prove that $\mathbb{R}$ is unique using its field definition? [on hold]

Let $({\mathbb{R}},+,*,≤)$ be a totally ordered Dedekind complete field. How to prove that it's unique without using other definitions of $\mathbb{R}$?
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1answer
22 views

Show that the set of all polynomials $f$ in $F[x]$ such that $f(A)=0$ is an ideal

Let $A$ be an $n \times n$ matrix over a field $F$. Show that the set of all polynomials $f$ in $F[x]$ such that $f(A)=0$ is an ideal. I don't understand how to apply this when it comes to ...
2
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1answer
42 views

If $v$ is a valuation and $v(x)<v(y)$ then $v(x+y)=v(y)$. [duplicate]

I was studying from some book and I came across something I haven't been able to justify. Let's suppose se have a field $K$ and an ordered group $G$ (with multiplicative notation) with an extra ...
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1answer
88 views

What kind of algebraic structure is $\left( \mathbb{R}_{\geq 0},+,\cdot \right)$?

Let $\left( \mathbb{R}_{\geq 0},+,\cdot \right)$ denote the non-negative real numbers with usual addition and usual multiplication. Obviously, this is not a field, because $0$ is the only additively ...
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1answer
52 views

Name of and references for the equivalence relation $x \sim y :\Longleftrightarrow x^2 = y^2$

Playing around with the concepts of negativity and positivity, I came across the following equivalence relation defined for all elements $x,y$ of a field $\mathbb{F}$: $$ x \sim y :\Longleftrightarrow ...
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0answers
92 views

I want to self study systematically pure mathematics? Where do I start? [on hold]

I am an undergraduate student in Mechanical Engineering and I am highly interested in studying pure mathematics systematically.I have a fair amount of knowledge on real and complex analysis, ...
0
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1answer
26 views

How can I show that $(x+1)$ and $(x^2+x+1)$ are irreducible in $\mathbb{R}$?

How can I show that $(x+1)$ and $(x^2+x+1)$ are irreducible in $\mathbb{R}$? For $(x+1)$, I'm not sure if it suffices to say $(x+1)$ has degree 1 so it is irreducible?
0
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2answers
60 views

Quaternion Multiplication: What is the correct way of doing it?

I am not very familiar with quaternions, I was just doing a programming homework were I had to implement quaternions' arithmetic, however I got puzzled by the multiplication of 2 quaternions. Let's ...
4
votes
2answers
157 views

When is a divisible group a power of the multiplicative group of an algebraically closed field?

It is known that for any algebraically closed field $\mathbb{F}$ its multiplicative group $\mathbb{F}^*$ is a divisible group, and consequently any power $\mathbb{F}^*\times\cdots\times \mathbb{F}^*.$ ...
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2answers
22 views

For complex polynomials $\gcd(f,g)=1$ if and only if $f$ and $g$ have no common root [on hold]

Assuming the fundamental theorem of algebra, prove the following. If $f$ and $g$ are polynomials over the field of complex numbers, then $\gcd(f,g)=1$ if and only if $f$ and $g$ have no common root.
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2answers
225 views

Is every submodule of a free module of finite rank over a PID a direct summand?

Suppose $F$ is a free module of rank $n$ over a PID, with $N$ a submodule. Is $N$ always a direct summand of $F$? I think the answer is yes, $N$ is also free of rank $m\leq n$ since we are working ...
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0answers
35 views

Nilradical of a polynomial ring

I am asked to compute the $nilrad(\mathbb{C}[X])$ and the reduction $\mathbb{C}[X]_{red}$. $\textbf{DEFINITION:}$ An element $a \in R$ is nilpotent if $a^n = 0$ for some positive integer $n$. ...
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1answer
20 views

Condition under which one can extend a linearly independent set in a free $\mathbb{Z}$-module to a basis?

I'm having difficulty answering the following simple (?) question: Let $X = \mathbb{Z}^n$, and let $M$ be a submodule of $X$ closed under division - by which I mean that if $x \in X$ has $n x \in M$ ...
6
votes
1answer
103 views

Tensor product of $\mathbb{Q}$ with an infinite product [duplicate]

How can I prove that the tensor $\mathbb{Q} \otimes \left( \prod_n \mathbb{Z}/n\mathbb{Z} \right)$, where the product is taken over all the positive integers $n$, is not trivial?
7
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3answers
165 views

G/N read as G modulo N.

In my abstract algebra course, the instructor is calling $G/N$ (the set of left Cosets of $N$ in $G$) $G\: mod\: N$. This has not yet been explained. Why is this the case? My immediate suspicion is ...
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0answers
26 views

Proof that an boolean algebra is free

I have a problem with proving that something is a free boolean algebra. I tried using the definition on the following problem: Prove that the boolean algebra of all the functions from a four element ...
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0answers
18 views

Prove free boolean algebra [on hold]

Prove that Boolean Algebra of all 4-elements subset 4-element set is free Boolean Algebra and has 2 free generators.
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2answers
68 views

Prove there is no simple group of order $729$

Let G be a group of order $729$. $729 = 3^6$ so by Sylow's Theorem G has a Sylow $3$-subgroup of order $729$. And there are $x$ of them. $r \equiv 1 \pmod 3$ and $r\ |\ 1$. So $x=1$? Is this ...
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2answers
45 views

Is $\mathbb R \times \mathbb Q$ a principal ideal domain

I have a similar question to this one: $\mathbb Z\times\mathbb Z$ is principal but is not a PID Is $\mathbb R \times \mathbb Q$ a principal ideal domain/ring (that is - is every ideal in $\mathbb R ...