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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Ideals in Algebra [closed]

Is there any geometric interpretation of ideals in algebra (like for instance one can intuitively imagine co-sets in terms of affine subspaces of a vector space)? Are there any instances in fields ...
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Smallest example of a group that is not isomorphic to a cyclic group, a direct product of cyclic groups or a semi direct product of cyclic groups.

What is the smallest example of a group that is not isomorphic to a cyclic group, a direct product of cyclic groups or a semi-direct product of cyclic groups? So finite abelian groups are ruled ...
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$O_{2n}(\mathbb{R}) \cap GL_{n}(\mathbb{C})=U(n)$

During a lecture of a Lie Algebras yesterday, the professor of the class stated the following fact without proof $O_{2n}(\mathbb{R}) \cap GL_{n}(\mathbb{C})=U(n)$ Note that we are viewing ...
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4answers
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Prove by definition that $(x,2)\subset\mathbb Z[x]$ is a maximal ideal

When the polynomial ring $\mathbb{Z}[x]$ is quotiented by the ideal $(2,x)$ we get a field as $\mathbb{Z}[x]/(x,2)\cong\mathbb{Z}/(2)\cong\mathbb{Z}_{2}$ which is a field. But I ...
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1answer
32 views

Generator operator of $v$?

Let $\text{U}_+$ be an associative $\mathbb{C}$-algebra with two generators $E$, $H$, and one defining relation $HE - EH = 2E$. Let $M$ be an $\text{U}_+$-module. If $v \in M$ is a nonzero eigenvector ...
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67 views

Irreducible Subrepresentations of representation of $\operatorname{GL}_{3}(\mathbb{F}_q)$

For a character $\zeta$ of $\mathbb{F}_q^*$, we can construct the representation $\zeta \otimes \zeta \otimes \zeta$ of the diagonal subgroup $L$ of $\operatorname{GL}_{3}(\mathbb{F}_q)$, in the ...
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21 views

Generalized Associative Property (Proof Verification)

I am really confused about Associative property and Generalized associative property. I am not sure of my proof, and I have a feeling that it is not correct. Would be happy if someone can tell me what ...
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What do we call collections of subsets of a monoid that satisfy these axioms?

Consider a monoid $M$ and a semiring $S$. Then there's an $S$-algebra freely generated by the monoid $M$, which can be described explicitly as the set of all finitely supported functions $S \leftarrow ...
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2answers
59 views

Prove that $\mathbb{Z}[i]$ consists precisely of the elements of $\mathbb{Q}(i)$ which satisfy $x^2 + ax + b=0$, $a,b \in \mathbb{Z}$

I was reading Neurkich's "Algebraic Number Theory" and there was a proof in it that makes no sense. Proposition 1.5: $\mathbb{Z}[i]$ consists precisely of the elements of the extension field ...
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minimal polynamial of $i\sqrt{5}+\sqrt{2} \in C$

Find the minimal polynamial of $i\sqrt{5}+\sqrt{2}\in C$ over rational numbers. My solution is; Say $\alpha=i\sqrt{5}+\sqrt{2}$ $(\alpha-\sqrt{2})^2=(i\sqrt{5})^2$ $\alpha^2-2\sqrt{2}\alpha=-5$ ...
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2answers
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Deducing factorization over $\mathbb{F}_p$ from factorization over $\mathbb{Q}$

I want to show rigorously that factorization over algebraic extensions of $\mathbb{Q}$ automatically yields a corresponding factorization over $\mathbb{F}_p.$ Consider for example the polynomial ...
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39 views

Find elements $x,y$ where $x\ne \pm1$ and $y\ne \pm 1$ in the field $\mathbb{Q}(\sqrt{5})$ satisfying $xy=19$.

Find elements $x,y$ where $x\ne \pm1$ and $y\ne \pm 1$ in the field $\mathbb{Q}(\sqrt{5})$ satisfying $xy=19$. I'm lost as to what to do. Any solutions or hints are greatly appreciated.
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40 views

How to find the number of solution of $x^n=1$ in the group $S_n$?

Suppose that $S_n$, the symmetric group of order $n!$ is given and for given $m\in \mathbb N$ fixed, we are to find the number of solutions to $\theta^m=e, \theta\in S_n$. Can someone tell me or give ...
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22 views

What is a conjugate weight?

The authors here write that the longest element of the Weyl group is $$w_{\max} = - id$$ except for $E_6$, $A_r$ and $D_r$ with $r$ even. There they write that $w_{\max}$ acts on a weight $\lambda$ ...
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On the number of group homomorphisms from $S_n$ to $S_m$

I was studying the number of group homomorphisms from $S_n$ to $S_m$ with $n\geq m\geq 7$ in this article. I have some difficulty in understanding. First of all, why such condition is given $n\geq ...
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1answer
48 views

Are all rings $\mathbb{Z}$-modules?

In my course of associative algebra we covered modules and an excercise involved showing that every ring $R$ can also be viewed as an $R$-module. This was straightforward enough. Is it also true that ...
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Inference of an identity in Grassmann algebra.

I am reading Herbert Federer's book called "Geometric Measure Theory", in chapter one of Grassmann algebra, on pages 36-37, he says that for $f$ being an endomorphism of a finite dimensional inner ...
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1answer
73 views

define two functions whose compositions are equal to identity

Let B be the set $B = \{1,2,....n\}$ where n is a positive integer. Let C be the set of all bitstrings of length n and let Z be the set of all functions from B to $\{0,1\}$. How do I find the two ...
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1answer
27 views

Check if algebraic structure is a field

Check if algebraic structure $(\mathbb{R^2},+,\cdot)$ is a field where binary operations $(+)$ and $(\cdot)$ are given by $$(x,y)+(u,v)=(x+u,y+v)$$ $$(x,y)\cdot(u,v)=(xu-2yv,xv+yu)$$ Structure ...
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If $A$ is a noncommutative ring then all biderivation is inner.

Before the question I will post some definitions: Derivation: An additive map $\delta: A \longrightarrow M$, where $A$ is a ring and $M$ is a $(A,A)-$bimodule is called derivation if $\delta(xy) = ...
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12 views

Does complexification make a self-conjugate representation non-self-conjugate?

I recently learned that a non-self-conjugate representation is not the same as a complex representation. Given a real representation $\pi$, with highest weight $\mu$ $$\pi : \mathfrak{g} \rightarrow ...
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A Problem for Nil-Ideals

Consider a ring $R$ and $I$ be a finitely generated nil-ideal of $R$. Is $I$ a nilpotent ideal? I have proved this for commutative rings. But for non-commutative rings I think this may not be true. ...
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gcd of $x$ and $2$ in $Z[x]$

In $Z[x]$, $x$ and $2$ has gcd $1$. But they cannot be expressed as the linear combination of two polynomials. Then assuming that $1=2.f(x)+x.g(x)$ we are supposed to arrive at a ...
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1answer
28 views

Every ring with $1$ and with no zero divisors and no non-trivial ideals is a division ring

It is well known that every commutative ring with unity $R$ that contains no non-trivial ideal is a field, since given $a \neq 0$, $(a)=R$, therefore there exists $x \in R$ with $ax=xa=1$. What ...
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2answers
34 views

Let $q$ be a prime integer. Show that for each $x∈GF(q)$ there exist elements $r$ and $s$ in $GF(q)$ satisfying $x=r^2+s^2$. [duplicate]

Let $q$ be a prime integer. Show that for each $x∈GF(q)$ there exist elements $r$ and $s$ in $GF(q)$ satisfying $x=r^2+s^2$. I'm stuck on this problem. Any solutions or hints are greatly appreciated. ...
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0answers
30 views

$M \otimes_\mathbb{C} N$ is simple.

Let $A$ and $B$ be finitely generated $\mathbb{C}$-algebras. For any simple modules $M$ and $N$, over $A$ and $B$ respectively, how do I see that the $A \otimes_\mathbb{C}B$-module $M ...
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38 views

Any two $A$-modules of the same dimension over $k$ isomorphic as $A$-modules?

Let $A$ be a central simple $k$-algebra. Are any two $A$-modules of the same dimension over $k$ isomorphic as $A$-modules?
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If $G$ is a Finite Group such that $H\le K$ or $K\le H$ for all Subgroups $H,K$ of $G$, then $G$ is Cyclic and of order $p^n$ for some Prime $p$.

Since each subgroup $K$ is contained in some other subgroup $H$, we can list the subgroups of $G$ in ascending order $$\lbrace 1 \rbrace < G_1< G_2 < G_3\cdots< G_{k-1}<G$$ By ...
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1answer
60 views

“Every element of Sym$(n)$ has order at most $n$”

I was doing mini-test involving a True/False section and came across the following statement. Every element of $Sym(n)$ has order at most $n$ I admit I had gotten this incorrect as I had thought ...
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2answers
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Normalizer of a Sylow 2-subgroups of dihedral groups

I can't solve the following exercise which is the last exercise in page 146 of Dummi & Foote's Abstract Algebra: Let $2n=2^ak$ where $k$ is odd. Prove that the number of Sylow 2-subgroups of ...
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1answer
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What is the injective envelope of a product of abelian groups?

We know that any $R$-module has an injective envelope. Matlis has shown that any injective module over a Noetherian $R$ decomposes as a direct sum of indecomposable injectives, and that these are the ...
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Homomorphisms of twisted modules

Let $A$ and $B$ be $R$-bimodules, and $\alpha$ an $R$-automorphism. Write ${}_1A_\alpha$ for the right-twisted $R$-bimodule with action $(r\cdot x\cdot s)\mapsto rx\alpha(s)$. Is it true that a ...
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1answer
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Criteria for monoid

I'm struggling to understand how to test for an identiy. Take, the finite group $(G,*) = \{1,2,3,4,6,8,12\}$ where * denotes the binary operation of greatest common divisor. It is my understanding ...
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1answer
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Why mentioning monic is important for gcd and lcm?

Let $F$ is a field and $F[x]$ be the polynomial ring over $F$. Now in the definition of the gcd or lcm of any two polynomials $g(x)$ and $f(x)$ it is mentioned that the ...
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question asked around a weird concept

I am struggling with these questions. I dont know what is meant by the term system of representatives. anybody knows about these things?
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Forming a Cayley table [closed]

How to form a Cayley table using $*$ as a binary operation on $P(A)$, where $A=\{1,2,3\}$. And the solution of main diagonal in the Cayley table.
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1answer
25 views

For $N, M \unlhd G$ relation between $MN/(M\cap N)$ and $N/(M\cap N)\times M/(M\cap N)$

Let $N, M \unlhd G$. Is $MN/(N\cap M)$ isomorphic to some subgroup of $$ N/(N\cap M) \times M/(N\cap M) $$ and how to prove this?
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1answer
26 views

A surjective endomorphism (of a Noetherian ring) is injective.

The problem is stated as follows: "Let $R$ be a Noetherian ring and $\theta$ be a ring homomorphism from $R$ to $R$. Show that if $\theta$ is surjective then it is also injective." Regardless of the ...
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1answer
26 views

Distinct elements of relations

R is the relation on Z (integers) given by xRy ( X is related to Y ) if 3 divides (x-y), what are the distinct elements of R?
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Is $\sqrt[3]{-1}=-1$?

I observe that if we claim that $\sqrt[3]{-1}=-1$, we reach a contradiction. Let's, indeed, suppose that $\sqrt[3]{-1}=-1$. Then, since the properties of powers are preserved, we have: ...
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1answer
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group theory dihedral group problem [closed]

I am stuck in this problem. plz give some suggestion
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Diophantine equation: $13^x+3=y^2$

$$13^x+3=y^2\iff \left(4+\sqrt{3}\right)^x\left(4-\sqrt{3}\right)^x=\left(y+\sqrt3\right)\left(y-\sqrt3\right)$$ $\gcd\left(y+\sqrt3, y-\sqrt3\right)=1$, therefore ...
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1answer
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Proving that R is a equivalence relation. [closed]

Let $R$ be a relation on $\mathbb Z$ given by $xRy$ iff $3|(x-y)$. How do we prove that R is an equivalence relation?
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1answer
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The Binary Operation [closed]

Let $A$ is equal to $\{ 1,2,3 \}$. Binary Operator defined as $\star$ on $\mathcal P (A)$ by $X\star Y$ is equal to $(X-Y) \cup (Y-X)$ and also equal to $X\triangle Y$, where: $X,Y$ is a subset of ...
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Number of elements of the group $SL_6(\mathbb{Z}/p^k\mathbb{Z})$

For $p$ a prime, what is the number of elements of the group $SL_6(\mathbb{Z}/p^k\mathbb{Z})$, $k \ge 1$? I can answer the $k=1$ case. For each element of $SL_6(\mathbb{Z}/p\mathbb{Z})$, there are ...
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a problem in homological algebra

For $C$ is abelian group satisfied $pC=0$ with p is a prime number and $G$ is abelian group. prove that $Ext_{Z}(C,G)\cong Hom(C,G/pG)$ I thought about this in 2 hours but couldn't prove it!
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102 views

Maps to quotient rings

If $R$ is a ring, $\mathfrak{a}$ is an ideal of $R$ and $S=A[x,y,z,\dots]$ where $A$ is a commutative ring, then is there a ring $S^\prime$ such that there is a bijection: ...
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1answer
58 views

Roots of $\Phi_{31}(x)$ as roots of unity.

Let $\Phi_{31}(x)$ be the $31$-cyclotomic polynomial. I want to show that $\Phi_{31}(x)$ is the product of six irreducible quintic factors in $\mathbb{F}_2$. I am running into difficulties ...
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2answers
114 views

A question about an infinite sequence of elementary row operations

Do there exist matrices $A$ and $B$ such that $B$ can be transformed into $A$ only if an infinite number of elementary row operations are performed on $B$? "What can we multiply the top equation by ...
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Composition of module homomorphisms and their kernels

Let $R$ be a principal ideal ring and let $I$ and $J$ be two ideals of $R$. Suppose $\phi: R \times R \rightarrow R \times R$ and $\psi: R \times R \rightarrow R \times R$ are two $R$-module ...