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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moufang loops and invertible real octonioms

Let Q be a Moufang loop of invertible real octonions. Which variety of Moufang loops is generated by Q? All of Moufang loops?
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42 views

Isomorphism between $R$ and its dual space

Let $R$ be a finite dimensional algebra over a field $K$. If $f$ is an $R$-module monomorphism from $R$ to the dual $K$-space $\operatorname{Hom}_K(R,K)$ why it is onto? Thanks!
4
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83 views

A non-UFD such that $a^2 \mid b^2$ does not lead to $a\mid b$

Is there any non-UFD that is a commutative ring such that $a^2 \mid b^2$ does not always lead to $a\mid b$? It would be preferable if examples are something that does not involve ...
4
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1answer
91 views

Prove $g^2 = e$ if there is a subgroup of index 2 that does not contain $g$ for every $g \in G$.

I'm having some trouble with this question from a practice exam. Let $G$ be finite group. Suppose for every $g \in G$ other than the identity element $e$, there is a subgroup $H \subset G$ of index ...
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1answer
32 views

Quotient set cardinal in $\mathbb{Z}_{12}$

In $\mathbb{Z}_{12}$ define the equivalence relation xRy if $x^2 = y^2$ Then what is the cardinal of the quotient set?
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29 views

Integral ring extensions and finitely generated as a module

Let $A \subset B \subset C $ be rings. Suppose that $A$ is Noetherian and $C$ is finitely generated as an $A$-algebra. I want to show that $C$ is finitely generated as a $B$-module $ \iff $ $C$ is ...
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1answer
80 views

How does $A_n$ look in Aut$(X)$?

Let me phrase my question precisely: Let $X=\{1,2,3,...,n\}$, $ \ S_n=\mbox{Sym}\{1,2,3,...,n\}$ be symmetric group on $n-$letters. Let $\mbox{Aut}(X)$ denote the automorphism group of $X$. We ...
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35 views

Showing the existence of sub fields of a finite field

For each divisor $m$ of $n$, $GF(p^n)$ has a unique sub field of order $p^m$ . The proof of this theorem in Gallian goes like this : Suppose that $m$ divides $n$. Then, since : $(p^n-1) = ...
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29 views

homology commutes with direct sum and product?

I'm looking at exercise 1.2.1 from Weibel's Intro to Homological Algebra. (I need to show that homology commutes with direct sum and direct product.) Is it possible to show that cokernels commute with ...
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1answer
29 views

Cardinal of the quotient set

Let $X = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$. In $X \times X$ define the equivalence $(a,b)\:\mathcal{R}\:(c,d)$ if $a+b=c+d$. Then what is the cardinal of the quotient set? I know that $|X| = 9$, so $| X ...
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30 views

Proving that the symmetric difference of sets is a group

I wanted to ask about this problem. My book states: Let the symmetric difference be defined as $A + B=(A\setminus B) \cup (B \setminus A)$. It proceeds to define a power set as $P_D= ...
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67 views

What happens if the coefficients of polynomials are not taken from a field of real numbers?

I saw in my abstract algebra textbook that defines the gcd of a polynomial over a field (i.e. the coefficients of the polynomial is taken from a field). My question is that what happens if the field ...
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1answer
46 views

Characterizing C* algebra generated by elements.

Let $A$ be a C*-algebra and $A_0\subset A$. Then it is known that the $\mathbb{C}$-algebra generated by $A_0$ (i.e. the intersection of all sub-$\mathbb{C}$-algebras containing it ) is just the ...
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3answers
98 views

Is division allowed in rings and fields?

Is division allowed in ring and field? The definition of ring I am using here does not require the presence of multiplicative inverse. I think in general, division is not a well-defined ...
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3answers
58 views

A number system that is not unique factorization domain

Can anyone present a number system that is not unique factorization domain and is a commutative ring? So I want the case that does not involve polynomials/monomials or some trivial cases.
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1answer
34 views

Number of terms containing primitive root of unity

It is well known that the degree of the n-th cyclotomic polynomial is $\varphi(n)$, where $\varphi$ is the Euler totient function. I define the ${minimal}$ sum to be of the form \begin{align} \xi_0 ...
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1answer
103 views

About automorphisms of commutative semigroups

Suppose that $M$ is a commutative monoid and that the product $P$ of $M$ and the nonnegative integers $\mathbb{N}$ with addition has no nontrivial automorphisms. The set $S$ of pairs $(m,n)$ in $P$ ...
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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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120 views

$x^p - 1$ always have a factor congruent to $1$ modulo $p$? [closed]

I was doing some group theory analysis and found the above statement. can you disprove it? I am not sure with my work, I am new with Group Theory. p is an odd prime [Editor's Comment] My ...
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1answer
98 views

Help in this notation in Fulton's Algebraic Curves book

I'm reading Fulton's Algebraic Curves book, I'm stuck in the following proposition (page 105): In fact, what I didn't understand is the following notation in the proof of this proposition: Why ...
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1answer
21 views

Finite Linearly Ordered Abelian Monoids

This question concerns the proper definition of the phrase "finite linearly ordered abelian monoids". The sequence A030453 of OEIS counts the number of "finite linearly ordered abelian monoids". The ...
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Normal Groups and Quotient Groups

These concepts are currently confusing me. My reading first defined a normal subgroup as one that is the kernel of a group homomorphism. Then it introduced the terms "left coset" and "right coset," ...
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2answers
31 views

Normal subgroups, direct product and monomorphism problem

Let $G$ be a group and let$H,K$ be normal subgroups of $G$. Let $\pi_H,\pi_K$ be the projections on $H$ and $K$ respectively. Show that the map $$f:G/(H \cap K) \to G/H \times G/K$$ defined as ...
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1answer
53 views

Proof that presheaf is a sheaf for Spec

Atiyah Macdonald define presheaf (chapter 3, exercise 23) on the base of $Spec(A)$, where $A$ is commutative ring with $1$, as follows $$ \mathfrak{F}(X_f) = A_f, $$ where $X_f$ is a basic open set ...
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1answer
46 views

What are the steps to calculate the number of elements in a quotient?

Let $X = \{0,1,2,3,4,5,6,7,8,9\}$ and $ Y = \{0,2,4,6,8,9\}$. In $P(X) =$ power set of $X $ define the following relation: $$A R B \Leftrightarrow A \setminus Y = B \setminus Y $$ Then, how many ...
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1answer
47 views

Prove that if $D$ is a finite integral domain, then $D$ is a field.

Please, check my answer to item "a" below and help me to solve item "b": Problem: Let $D$ be an integral domain and consider $a \in D$; $a \neq 0$. a) Show that the function $\phi_a:D \rightarrow ...
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1answer
73 views

How do we know the classification of finite simple groups is finished?

I don't have much experience in group theory beyond knowing what a finite simple group is, as well as the other basic parts of the topic, so far as I can tell, so if the actual answer is too complex ...
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2answers
72 views

Idempotent elements of a ring.

I need the idempotent elements of $Z_{900}$ $2^2\cdot 3^2\cdot 5^2=900$ Of course there's $$0 \pmod 4 \\ 0 \pmod 9 \\ 0 \pmod {25} \\ $$ and $$ 1 \pmod 4 \\ 1 \pmod 9 \\ 1 \pmod {25} \\ $$ I found ...
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1answer
36 views

Dimension of an algebra/vector space

Does the dimension of an algebra/vector space have any connection to Euclidian spacial dimensions, for all algebras/vector spaces? I know some algebras/vector spaces can be represented in ...
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1answer
36 views

Question concerning the GAP package qpa (find all ideals with a certain property)

I have the following question concerning the GAP package qpa. Let $k$ be a fixed finite field and let $Q$ be a fixed quiver. Let $kQ$ denote the associated path algebra. Since $k$ is ...
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1answer
54 views

Order of group $GL_{2}\left( \mathbb{F}_{p}\right) $

I'm having a hard time counting. I need to count the number of elements for the multiplicative group of invertible $2\times 2$ matrices $GL_{2}\left( \mathbb{F}_{p}\right) $ with elements from the ...
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3answers
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In $\mathbb{Z}/(n)$, does $(a) = (b)$ imply that $a$ and $b$ are associates?

[Update: Based on the hints provided by @zcn and @whacka, I believe I have found a solution. See my answer below.] Below, $R$ is a commutative ring with $1$. In John J. Watkins' Topics in ...
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2answers
103 views

Generators of the Relations of a Galois Extension

Let $K$ be a Galois extension of $\mathbb{Q}$ of degree $n$. Pick some primitive element and take the roots $a_1, ..., a_n$ of its minimal polynomial. Then the evaluation map $\mathbb{Q}[x_1, ..., ...
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1answer
64 views

Proving Finiteness of Group from Presentation

Given the group $G = \langle a, b, c : a^2 = b^3 = c^5 = abc\rangle$, I want to show that $H = G / \langle abc\rangle$ is a finite group. I tried to find a canonical form for elements of $H$. That ...
5
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1answer
170 views

Sigma-Algebra: Is it an Algebra, Field, or Something Else?

The Wikipedia page for $\sigma$-algebra says this set is called a "sigma-algebra" by some, and called a "sigma-field" by others. I'm writing a paper on measure theory, where the topic of sigma-algebra ...
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1answer
116 views

Endomorphisms of the maximal ideal of a local ring

Let $R$ be a commutative local ring with maximal ideal $\mathfrak{m}$. Is it true in general that $\text{Hom}_R(\mathfrak{m},\mathfrak{m})\cong \text{Hom}_R(\mathfrak{m}, R)$? What if the Krull ...
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1answer
42 views

generators vs basis of an algebra

Are all bases of an algebra generating sets, but all generating sets are not bases? A basis can only use addition and scalar multiplication to generate an algebra, which means it is a generating ...
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Definition - Logarithmic Zeros and Algebraic Vector Fields

In Algebraic Geometry, what is the difference between an algebraic vector field and a derivation (are they completely different or synonymous)? What does it mean to say an algebraic vector field has ...
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2answers
47 views

Map from $\mathbb{C}^2$ to $\mathbb{A}_{\mathbb{Q}}^2$

I was doing Exercise 3.2 I on Ravi Vakil's notes on Algebraic Geometry: consider the map of sets $\phi: \mathbb{C}^2 \rightarrow \mathbb{A}_{\mathbb{Q}}^2$ defined as follows. $(z_1, z_2)$ is sent to ...
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3answers
49 views

When is a non-trivial homomorphism injective?

I noticed that over the natural numers $(\mathbb{Z},+)$ any group homomorphism $f : \mathbb{Z} \rightarrow \mathbb{Z}$ that is not the trivial one, is automatically injective. Where exactly does ...
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1answer
30 views

When a given ideal is a radical ideal

I am wondering if there are any canonical methods for checking whether a given ideal is radical. For example, I got stuck on the following example: Let $f=x+2y-z$ and $g=z-2w$ and let $I$ and $J$ be ...
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2answers
45 views

Proving that $f(x)$ is irreducible over $F(b)$ if and only if $g(x)$ is irreducible over $F(a)$

Let $f(x)$ and $g(x)$ be irreducible polynomials over a field $F$ and let $a,b \in E$ where $E$ is some extension of $F$. If $a$ is a zero of $f(x)$ and $b$ is a zero of $g(x)$, show that $f(x)$ is ...
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40 views

Least power of $x$ so that $y$ divides $x$

How do I find the least $z$ to satisfy, $$y \mid x^z$$ I have tried keep dividing $y$ with the GCD($x$,$y$) until $y \mid x$ and adding $z$ by $1$ (starting from $1$), but turns out it's too slow.
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Show that the kernel of the map $SL(n, \mathbb{Z}) \to SL(n, \mathbb{Z}/3\mathbb{Z})$ has no torsion.

I am trying to show that the kernel of the natural map $SL(n, \mathbb{Z}) \to SL(n, \mathbb{Z}/3\mathbb{Z})$ has no torsion. That is, if $A$ is in the kernel then $A = I$ or $A^n \neq I$ for all $n ...
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Element of a Finite Galois Group

With $f(x)$ being an irreducible, separable polynomial of finite degree and $G/L$ the Galois extension of $f(x)$. I know that $\exists x\in Gal(G/L)$ such that for all pairs of roots $(\alpha,\beta)$ ...
4
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2answers
46 views

Union of subgroups is subgroup

I am doing an exercise where I am asked to prove or disprove the statement: If $G$ is a group, and $H_1,H_2,H_3$ and $H_1 \cup H_2 \cup H_3$ are subgroups of $G$ then $\exists i,j$ with $i \neq j$ ...
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1answer
27 views

Implementing trig functions for dual numbers

I'm curious, how do common trig functions get implemented for dual numbers? One way would be to use the power series definition, but that seems inefficient
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33 views

Some wedge product computation

I want to calculate $w\wedge w$ for $w=\sum a_{ij} e_i\wedge e_j$ where the sum is over all $1\leqslant i<j\leqslant 4$. Is there a neat way to do it without writing all the terms out? What about ...
4
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2answers
106 views

Usage of the term “Free”

What is the difference between free groups and free modules/vector/algebras spaces, or in other words what does free mean in algebra? I have seen two uses of the term free: something of ...
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2answers
100 views

Galois theory, had it solved any major problems beside its original applications to classical problems?

Galois theory, had it solved any major problems beside its original (classical) applications to roots of a fifth (or higher) degree polynomial equation (solvable algebraic equations and constructible ...