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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3answers
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Index of intersection of two normal subgroups

Question: Let $G$ be a finite group and $H$ and $K$ are normal subgroup of $G$. If $[G:H]=2$ and $[G:K]=3$, determine $[G:H∩K]$. My attempt: I think $[G:H∩K] = 6$. Since $G$, $H$, $K$ are ...
0
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1answer
35 views

minimal subgroup is a direct product of simple groups

On page 112 of Dixon and Mortimer's Permutation Groups, Theoerm 4.3A (iii) says that every minimal normal subgroup $K$ of $G$ is a direct product $K=T_1 \times \cdots \times T_k$ where $T_i$ are ...
1
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0answers
30 views

Alternating form from hermitian product

Let $V_\mathbb{C}$ be a complex vector space and $h$ an hermitian product on it. In particular let $\{e_1,\dots,e_n\}$ be a base of $V_\mathbb{C}$, then $h:=\sum_{i,j=1}^nh_{ij}dz_id\overline{z}_j$ ...
-1
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0answers
38 views

irreducibility of monic polynomials over Z [closed]

Statement : Monic polynomials irreducible over Q are irreducible over Z. Where the polynomials belong to Z[x]. How to prove or disprove the statement. It seems like the converse of gauss lemma ...
0
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1answer
41 views

Is it the case that every UFD is noetherian on principal ideals?

Its not the case that every UFD is noetherian; the standard counterexample is $R[x_0,x_1,x_2,\ldots]$, which has the following ascending sequence of ideals: $$\langle \rangle,\langle x_0\rangle,\...
2
votes
1answer
18 views

Ideal generated by two irreducible polynomials is the field itself

The question is: Let $F$ be a field and $f(x),g(x) \in F[x]$. Verify that $$N=\{r(x)\ f(x)+s(x)\ g(x):r(x),s(x)\in F[x]\}$$ is an ideal of $F[x]$. Then show that if $f(x)$ and $g(x)$ have different ...
2
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2answers
55 views

Dihedral groups: Relationship between symmetries and rigid motions

In Dummit & Foote, $D_{2n},\ n \geqslant 3$ is the set of symmetries of a regular $n$-gon, where a symmetry is a rigid motion of the $n$-gon which can be effected by taking a copy of the $n$-gon, ...
1
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1answer
63 views

Any unit has an irreducible decomposition

The proposition is the following: Let $R$ be a principal ideal domain. Then every $a \in R$ with $a \neq 0$ has an irreducible decomposition, that is, there is a unit $u$ and irreducible elements $...
1
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0answers
55 views

Is there a diffeomorphism between any algebraic closure of $\Bbb{F_2}$ and some subset of $\Bbb{C}$?

I'm looking for a way to translate between differential equations and their solutions over some algebraic closure of $\Bbb{F_2}$ and some subset of $\Bbb{C}$. The algebraic closure of $\Bbb{F_2}$ ...
5
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3answers
96 views

Let $R$ be a commutative ring, $\phi :R\to S^{-1}R, \phi(r)=\frac{r}{1}$ then $\phi(r)$ is invertible iff $r\in S$

$R$ is an arbitrary commutative ring with identity, and $S\subset R$ is multiplicative. I read that the map $\phi :R\to S^{-1}R, \phi(r)=\frac{r}{1}$ is characterized by the set $S'=\{s:\phi(s)\text{ ...
0
votes
4answers
60 views

Explaining the definition about greatest common divisor, what do they mean up to the equivalence relation?

The definition of gcd is provided below: Let R be an principal ideal domain. Let a,b ∈ R. Then there is a greatest common divisor of a and b, that is, an element d that divides both a and b and such ...
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2answers
86 views

What does $\mathbb{Q}(\sqrt{2},\sqrt{3})$ mean? [duplicate]

What set is $\mathbb{Q}(\sqrt{2},\sqrt{3})$? Is it the set $X = \{a\sqrt{2}+b\sqrt{3}:a,b\in\mathbb{Q}\}$?
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2answers
47 views

How can $ \langle -1 \rangle $ be the same as C2?

How can $ \langle -1 \rangle $ be the same as $ \text{C2} =\{-1, 1\} $? Why can we just write $ \langle -1 \rangle $? Why do we say that $ \{-1, 1 \} $ is generated by $ \langle -1 \rangle $? With $ \...
2
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0answers
28 views

Nonunital C*-Algebra: Proper Ideals

Given a C*-algebra without unit. Does there exist a nontrivial proper ideal that does not lie in a maximally nontrivial proper ideal? (For the unital case this follows easily by Zorn's lemma.) ...
2
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1answer
31 views

Is every element of a finite dimensional commutative non-unital $\mathbb{R}$-algebra nilpotent?

Considering a few examples of finite dimensional non-unital algebras over the reals, I tried coming up with an example of such an algebra with non-nilpotent zero divisor elements. In all the examples ...
3
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1answer
69 views

On graded Artinian Gorenstein algebras

Let $k$ be a field and $R$ an $\mathbb{N}$-graded $k$-algebra that is graded-commutative. Assume that $\dim_k R<\infty$ and that $R$ is Gorenstein (i.e. the injective dimension of $R$ over itself ...
0
votes
1answer
39 views

Decompose and compute the sign of $\sigma(k)=n+1-k$

Let $n\geq 2$ and $\sigma$ is permutationof $\{1,2,\ldots,n \}$ defined by : $$\sigma(k)=n+1-k$$ Decompose permutation $\sigma$ into product of disjoint transpositions and compute the sign of it ?...
0
votes
1answer
60 views

How to prove that $\langle\{ (1,2),(1,2,\ldots,n) \}\rangle=\mathfrak{S}_n$

Let $n\geq 2, \tau=(1,2),\ c=(1,2,\ldots,n)$ two permutation of $\mathfrak{S}_n$ Prove that $$\biggl\langle\{ (1,2),(1,2,\ldots,n) \}\biggr\rangle=\mathfrak{S}_n$$ Indeed, normally i will ...
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0answers
2 views

Turing machine with k-dimensional tape or k-regular tree

The statement I read is " In a k-dimensional tape, cells corresponds to elements of free commutative group of k generators. s. There are 2k shifts, which correspond to addition of a generator ...
1
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1answer
44 views

On boolean algebras as rings, modules and/or R-algebras

After trying to make sense of first order logic from an algebraic point of view I started to read about boolean algebras (similar to the explanations given here: wikipedia on boolean algebras. I also ...
0
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2answers
60 views

Help with understanding quotient ring structure

Let $R$ be the ring $\mathbb{Z}[x]/((x^2+x+1)(x^3+x+1))$ and $I$ be the ideal generated by $2$ in $R$. What is the cardinality of the ring $R/I$? I am having a hard time understanding what the ring ...
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2answers
21 views

Definition of irreducible element of a ring

I found in my notes the following definition: Let $r\neq 0$, $r$ non-invertible. $r\in R$ is called irreducible iff $r=a\cdot b$ with $a,b\in R$ then either $a\in U(R)$ or $b\in U(R)$. Why does ...
1
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1answer
41 views

Proving $\Bbb Z/p\Bbb Z\cong R/pR$

Let $R$ be a ring of square-free order $n$. If $p \mid n$ then $\Bbb Z/p\Bbb Z\to R/pR$ is a well-defined isomorphism. I'm really unsure how to approach this problem. So we need to show that if $...
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2answers
21 views

Element separated from a submodule by a homomorphism

Let $M$ be a module over a commutative ring $A$, $M'$ a submodule, and $y\in M\setminus M'$. Then $y$ can be separated from $M'$ by homomorphism. What I wish to prove is that there exists an $A$-...
0
votes
1answer
42 views

Certain Subset of a Ring [closed]

Does there exist an infinite ring $R$ with finitely many units and an infinite $S \subseteq R$ \ $\left\{0 \right\}$ such that: There exists finite $X \subseteq S$ such that for every $y \in S$, ...
0
votes
0answers
101 views

The endomorphisms of an $A_\infty$-algebra as a (bi)module over itself

Let $A$ be a unital associative algebra. A well-known exercise states that the ring of $A$-bimodule endomorphisms of $A$ are isomorphic to the center of $A$. That is, $\text{End}_{A-\text{mod}}(A) \...
2
votes
2answers
56 views

Number of solutions a polynomial can have as a function of the field?

Is there any limitation (upper bound) for number of solutions of polynomial equations? Having a background in engineering, my knowledge of higher algebra is rather limited, but I do know of ...
2
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2answers
29 views

Generating a subfield with identity element

I know this is a very basic question but I am always unsure what exactly is meant by "generating". For example, consider the polynomial ring $k[x]$, and the ideal generated by $f(x)$, denoted by $\...
1
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2answers
49 views

Prove that a normal subgroup $K$ is characteristic in a finite group $G$ if $\gcd(|K|, |G/K|) = 1$

This is a part of an exercise $2.2$ on a page $203$ in a book "Algebra: Chapter 0" by P.Aluffi. Let $K$ be a normal subgroup of a finite group $G$, and assume that $|K|$ and $|G/K|$ are relatively ...
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1answer
25 views

Counting the number of distinct elements in Sylow subgroups if $|G|=30$

I'm trying to prove that if $|G|=30=2\cdot 3\cdot 5$ then $G$ has a normal $3$-Sylow or a normal $5$-Sylow. By the Sylow theorems, we would argue by contradiction in order to prove it cannot be $n_3=...
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0answers
77 views

Trigonometric solution to solvable equations

The algebraic equations in one variable, in the general case, cannot be solved by radicals. While the basic operations and root extraction applied to the coefficients of the equations of degree $ 2 $ ,...
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0answers
43 views

Action of a Linear Functional on a Polynomial

I was hoping to find a good canonical reference for the mathematics behind something called the action of a linear functional $L$ on a polynomial $p(x)$ which is denoted $\langle L|p(x)\rangle$ ...
7
votes
4answers
407 views

Is my understanding of quotient rings correct?

Amidst all the rigorous constructions of quotient rings involving equivalence relations and ideals, I feel that I have finally grasped what a quotient ring is. I have applied this intuition to a few ...
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0answers
51 views

Question about the Fundamental Theorem of Galois Theory

This is something more like a small doubt than some problem that I need help with. I'm doing and exercise that is asking me to find subextensions of a given extension of all posible orders, and find ...
0
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1answer
67 views

Is $m\mathbb{Z}\otimes_{\mathbb{Z}}\mathbb{Z}/m\mathbb{Z}\cong 0$?

Since each 'generator' of $m\mathbb{Z}\otimes_{\mathbb{Z}}\mathbb{Z}/m\mathbb{Z}$ has the form $km\otimes_{\mathbb{Z}}\bar{a}=k\otimes_{\mathbb{Z}}m\bar{a}=k\otimes_{\mathbb{Z}}0=0$.
58
votes
7answers
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Does associativity imply commutativity?

I used to think that commutativity and associativity are two distinct properties. But recently, I started thinking of something which has troubled this idea: $$(1+1)+1 = 1+ (1+1)\implies 2+1=1+2$$ ...
0
votes
1answer
19 views

what is inductive class of algebras?

When I was studying the following lemma from an article , I faced to some notations I was not familiar with them, I would appreciate your help to find out them: Lemma:Let $X$ be an inductive ...
0
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0answers
29 views

Taylor's expansion in polynomial ring

While looking proof of Hilbert's Nullstelllensatz in M. Artin's Algebra, I faced some problems. See below: The last expression for $f(x)$ is called in the book as Taylor's expansion. ...
2
votes
2answers
35 views

Does element of the same conjugacy class commute?

Let $G$ a group and $C_1,...,C_n$ it's conjugacy class. I have to compute $[G,G]=\left<ghg^{-1}h^{-1}\mid g,h\in G\right>$. I would like to know if elements of $C_i$ commute. I know that element ...
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votes
2answers
32 views

Showing the kernel of $f:\mathbb{Z} \rightarrow \mathbb{Z}_n, f(x)=[x]_n$ is the ideal $n\mathbb{Z}$? [closed]

How to display that the kernel of $f:\mathbb{Z} \rightarrow \mathbb{Z}_n, f(x)=[x]_n$ is the ideal $n\mathbb{Z}$? I cannot understand from reading "the kernel is $n\mathbb{Z}$" that it really is the ...
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1answer
98 views

Basic Algebras: Definition

A $k$-algebra $A$ is called basic if for every set of primitive orthogonal idempotents $\left\{e_1, \dots , e_n\right\}$ such that $1=\sum_{i=1}^ne_i$ we have that $$e_iA\cong e_jA\Leftrightarrow i=j.$...
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0answers
30 views

Books about multivariate polynomials

I'm looking for a book on multivariate polynomials, preferably a monograph (could also be a chapter inside another book). I'm interested in what can be said about roots, factoring, irreducibility, ...
3
votes
2answers
69 views

Ring of order $n$ is isomorphic to $\Bbb Z/n\Bbb Z$, with $n$ square-free

Let $R$ be a ring of order $n$ and suppose $n$ has no square in its prime decomposition. How do I see that $R$ is isomorphic to $\Bbb Z/n\Bbb Z$? I bet that the map $\Bbb Z \to R, \, 1\mapsto 1_R$ ...
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2answers
67 views

Ring with four solutions to $x^2-1=0$

I am looking for a ring $R$ in which $2$ is invertible and there are four solutions to $x^2-1=0$. $R=\Bbb Z/8\Bbb Z$ has the four solutions $1,3,5,7$ to $x^2-1=0$, but $2$ is not invertible.
2
votes
1answer
53 views

Conjugacy classes of $\mathcal D_{10}$.

I was wondering if there is a special technique to find the conjugacy classes of $\mathcal D_{10}=\left<a,b\mid a^5=b^2=1,bab^{-1}=a^{-1}\right>$, and of $\mathcal D_{2n}=\left<a,b\mid a^n=b^...
2
votes
1answer
36 views

Socles and factors

Let $A$ be a finite dimensional algebra over a field $K$ and let $M$, $M'$ and $N$ be $A$-modules. Suppose that $M'\subseteq M$ and assume that $N$ has simple socle. Let $f: M \longrightarrow N$ be ...
3
votes
1answer
49 views

Nilpotence and conjugacy in $M(p,\mathbb F_p)$

I have to solve the following problem: Characterize matrices $X\in M(p,\mathbb F_p)$ (note that $p$ is the dimension and the characteristic of the field) such that there exists $Y$ with the ...
0
votes
1answer
35 views

A reduction to the finite degree case

I am stuck trying to understand a proof in Asymptotic Differential Algebra and Model Theory of Transseries by L. van den Dries, J. van der Hoeven and M. Aschenbrenner. The result is the following: ...
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2answers
89 views

Let $G$ be a group and $N$ be a minimal normal subgroup of $G$, $G_p \in Syl_p(G)$, where $p$ is an odd prime divisor of $|G|$.

Let $G$ be a group and $N$ be a minimal normal subgroup of $G$, $G_p \in Syl_p(G)$, where $p$ is an odd prime divisor of $|G|$. Suppose that $M/N$ be a maximal subgroup of $G_pN/N$. then $M = PN$ for ...
1
vote
1answer
32 views

On the definition of free product of groups.

Let $G$ and $H$ be groups. Their free product is a group $G*H$ with homomorphisms $\iota_G:G\rightarrow G*H$ and $\iota_H:H\rightarrow G*H$ so that given any other group $X$ with homomorphisms $f_G:G\...