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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Understanding linearly independent vectors modulo $W$

We've learned in class: Let $W \subseteq V$, a subspace. $v_1, \ldots, v_k \in V$ are said to be linearly independent modulo $W$ if for all $\alpha_1, \ldots, \alpha_k: \sum_{k=1}^n \alpha_i v_i ...
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Is there an accepted name for algebraic structures like $\mathbb{Q}_{>0}$ and $\mathbb{R}_{>0}$?

Question. Is there an accepted name for algebraic structures that, like $\mathbb{Q}_{>0}$ and $\mathbb{R}_{>0}$, are models of the algebraic theory presented as follows? Sorts: $U$ Functions: ...
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Integral domain - Embedding

Let $R$ be an integral domain and the homomorphism \begin{align} \phi\colon \mathbb{Z} &\rightarrow R \\ n &\mapsto n \cdot 1_R \end{align} What does it mean that if $\ker \phi =\{0\}$ then ...
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$A$ integral domain implies $A[x]$ integral domain (proof check)

I'm studying for my abstract algebra course and want to prove as an exercise that if $A$ is an integral domain then $A[x]$ is an integral domain. I realized later that there is a more direct proof, ...
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0answers
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Computer program for factorization into irreducible polynomials over $\mathbb{Z}_p^k$

Hensel's Lemma allows us to factor a polynomial uniquely into basic irreducible factors over $\mathbb{Z}_p^k$. Is there a SAGE or Magma command that gives this factorization? Or can anyone help in ...
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1answer
12 views

Concerning Adam's theorem/binary continous operation on Spheres

First I will quote Adam's theorem (also known as the Hopf invariant theorem) from here: "The Hopf invariant one theorem, sometimes also called Adams' theorem, is a deep theorem in homotopy theory ...
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2answers
25 views

Non-singular matrices forms a group

Do the set of all non-singular matrices forms a group under multiplication. ? I don't think it does as the multiplication operator is not even defined for every two element in the set . Just need a ...
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1answer
12 views

Suppose $R=F$ is a field. Prove that an $R-$module $M$ is Artinian iff it's Noetherian iff $M$ is a finite dimensional vector space over $F$.

Suppose $R=F$ is a field. Prove that an $R-$module $M$ is Artinian iff it's Noetherian iff $M$ is a finite dimensional vector space over $F$. If $M$ is a finite vector space over $F$, then neither do ...
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31 views

Projective resolution of k over R=k[x,y]/(xy)

I want to prove that $\operatorname{Tor}_{n}^{R}(k,k)=k\oplus k,\,\,\forall n\ge 1$. I found the projective resolution $$ R^4\longrightarrow R^3\longrightarrow R^2\longrightarrow R ...
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Endomorphisms of a ring

Let $R$ be a ring with identity and let $R^n=P⊕P'$ be a direct sum decomposition with right $R$-modules as its components. We take $e\in End(R^n_R)$ as the projection of $R^n$ onto $P$, namely, ...
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51 views

What is the algebraic role of the mathematical constant $\gamma$?

Mathematical constants $\pi$, $e$, $i$ have a lot of algebraic roles. They appear as identity elements, idempotents, invariant elements etc against various operations and sets. This is illustrated by ...
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$P$ is normal complement in $G$ if and only if $H$ has normal subgroup $Q$ which $H=P\times Q$.

suppose that $G$ is finite group and $P$ is aabelian $p$-sylow subgroup of $G$ and $H=N_{G}(P)$. show that $P$ is normal complement in $G$ if and only if $H$ has normal subgroup $Q$ which $H=P\times ...
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1answer
32 views

Finite groups and Normal subgroups - $G$ is a finite group such that $|G|=n$. Let $p$ be the smallest prime such that $p|n$.

$G$ is a finite group such that $|G|=n$. Let $p$ be the smallest prime such that $p|n$. If $H\triangleleft G$ such that $|H|=p$, then prove that $H\subset Z(G)$. My Work: Since $|H|$ is a prime ...
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Galois group of $x^8-2$ and intermediate fields

Let $p(x)=x^8-2$ be a polynomial over $\mathbb{Q}$. I am to find its Galois group $G$ and the correspondence between subfields of its splitting field $\mathbb{K}_p$ and subgroups of $G$. It is easy to ...
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3answers
116 views

Unit Ideal and its generators

Let $R$ be a commutative ring with unit and let $a,b\in R$ be two elements which together generates the unit ideal. Show that $a^2$ and $b^2$ also generate the unit ideal together. My Work: Unit ...
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1answer
41 views

Automorphisms in $\mathbb{R}$

Let $\phi: \mathbb{R}\rightarrow \mathbb{R}$ be an automorphism. Suppose $p=\frac{m}{n}$ is a rational number. Then is it true that $\phi(p)=\frac{\phi(m)}{\phi(n)}$? I got this problem while doing ...
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22 views

A bimodule homomorphism

$\def\Hom{\operatorname{Hom}}$ Let $R$ be a ring and $P_R$ be a right $R$-module. Set $Q=\Hom_R(P,R)$ and $S=\Hom_R(P,P)$, both operating on the left of $P$. This makes $P$ into an $(S,R)$-bimodule. ...
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Solving systems of polynomials with an oracle

I need to solve a system of polynomials. Let the variables be $x_1, \dots, x_n$, and let the polynomials be $f_1, \dots, f_n$ Let's say we have these conditions we can already assume: there are ...
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43 views

homomorphisms of finitely generated abelian groups [on hold]

If $A$ and $B$ are finitely generated abelian groups such that $\text{Hom}_{\mathbb{Z}}(A,B)\neq0$ and $\text{Hom}_{\mathbb{Z}}(B,A)=0$, I want to prove that $B \otimes_{\mathbb{Z}} \mathbb{Q}=0$ and ...
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1answer
46 views

How many elements of order 5 may contain in a group of order 90? [on hold]

G - group of order 90 THE QUESTION: How to found the count of elements of order 5 in this group?
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3answers
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Finding all abelian groups such that there exists certain short exact sequence.

I have to find all abelian groups $A$ such that there exists a short exact sequence $0\rightarrow\mathbb{Z}\rightarrow A\rightarrow\mathbb{Z}\oplus\mathbb{Z}_{6}\rightarrow 0$. I have found ...
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Problems from the Kourovka Notebook that undergraduate students can fully appreciate

The Kourovka Notebook is a collection of open problems in Group Theory. My question is: could you point out some (a "[big-list]" of) problems [by ...
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Number of $p$-local subgroups of a group

A subgroup of a finite group is '$p$-local' if it is the normalizer of some Sylow $p$-subgroup. I want to prove that the number of $p$-local subgroups of a group is congruent to $1$ modulo $p$. I know ...
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1answer
41 views

Integral closure of a PID is torsion free

Can anyone explain me why the integral closure of a PID $A$ in a separable finite extension of its fraction field is a torsion free $A$-module? I know that it is a finitely generated A-module ...
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1answer
75 views

Equivalent conditions for an ideal to be prime

Let $R$ be a commutative ring. An ideal $I$ is called prime if whenever $ab\in I$ then $a\in I$ or $b\in I$. I want to show that $I$ is prime if whenever $JK\subseteq I$, then $J\subseteq I$ or ...
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1answer
19 views

Negative degree valuation: valuation ring and its maximal ideal

I know that the $v: f \mapsto -\deg(f)$ is a discrete valuation on the field of complex rational functions $\mathbb{C}(X)$ (the quotient field of $\mathbb{C}[X]$). The valuation ring $\mathcal{O}_v$ ...
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1answer
69 views

Does $(X)(Y)=(XY)$ for $X,Y\subseteq R$?

Let $R$ be a commutative ring. Denote by $X\ast Y=\{xy\mid x\in X,y\in Y\}$ the complex product of subsets. I want to show that given subsets $X,Y\subseteq R$ the following ideals are equal: ...
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25 views

Galois Ring and its maximal cyclic subgroups [on hold]

Please let me know about the theory of Galois Ring. Also me let me know how to calculate Maximal cyclic subgroup of degree 15 in Galois Ring.
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Factoring a polynomial over $\mathbb F_{2^8}$

How do you find the factors of $x^4+x+1$ in $GF(2^8)$ in terms of polynomials? Let me explain, We have primitive irreducible polynomial $p(x)=x^2+x+1$ in $GF(2^2)$ which has root $\alpha^2+\alpha$ in ...
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1answer
28 views

Exponents of the Coxeter group A(n)

I came across the following result : "The exponents of the Coxeter group A(n) (n>=1) are 1, 2, ... , n." I am not able to figure out a proof of this fact. Any help towards proving this result will be ...
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1answer
80 views

Finite integral domains are commutative?

Here, integral domain is a non-zero ring $R$ (not necessarily commutative, and not necessarily contains unity), in which $ab=0$ implies $a=0$ or $b=0$. Question If $R$ is a finite integral domain, is ...
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1answer
45 views

Prove that for all $x \in R$, the ideal $xR$ is proper.

Let $R$ be a commutative ring without identity. Suppose $R$ doesn't contain a proper maximal ideal, and $R$ is not the zero ring. Prove that $\forall x \in R$, the ideal $xR$ is proper.
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Adjoining a number to a field

When I studied algebra, we talked about fields such as $\mathbb{Q}[\sqrt{2}]$, the rational numbers with the square root of two adjoined to the field. Structures like these are called field extensions ...
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2answers
34 views

$G$ contains at least $r(p-1)$ elements of order $p$

Suppose a group $G$ has $r$ distinct subgroups of prime order $p$. Show that $G$ contains at least $r(p-1)$ elements of order $p$. Aside: I know how to use this to prove that a group of order $56$ ...
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1answer
35 views

HK is cyclic if H and K are both cyclic

Let $G$ be a finite group with normal subgroups $H$ and $K$ of relatively prime orders. Show that the group $HK$ is cyclic if $H$ and $K$ are both cyclic. My attempt was to use the $2$nd Isomorphism ...
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2answers
49 views

Symmetric Group Action on a Set of Functions

I'm not understanding the following passage and I'm hoping someone could elucidate (for context, this is in the lead up to the definition of the sign of a permutation) where the author says: Let ...
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Transcendence degree of polynomials equals the transcendence degree of the function field of the polynomials

How to prove that transcendence degree of the function field $k(f,g)$ is equal to the transcendence degree of $\{f,g\}$ where $f$ and $g$ are two multivariate polynomials? If $\{f,g\}$ are ...
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1answer
45 views

Why are minimal irreducible closed sets in $A^n$ single points?

In Hartshorne's Algebraic Geometry example 1.4.4, he says A maximal ideal $m$ of $A = k[x_1,\cdots,x_n]$ corresponds to a minimal irreducible closed subset of $A^n$, which must be a point ... I ...
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Group of Order 33 is Always Cyclic

I would love to get help on this problem from a chapter on Commutator of Group Theory: Show that each group of order 33 is cyclic. (Hint: Use the result from the Exercise and Lemma below.) ...
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1answer
70 views

A question about a free abelian finitely generated group.

I am having a hard time solving this and it is really confusing. I don't have enough schema, which makes it problematic. Let $A$ be a finitely generated free abelian group and $B$ is a subgroup of ...
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1answer
40 views

Finding suitable basis for a free abelian finitely generated group.

I am stuck with this exercise forever... I was barely taught about it, English is not my mother language and in any other phrasing it is not coherent with my material.I'd really appreciate your help. ...
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Is there a convention, law or axiom for associate operators when is a lack of brackets?

If a have an operator $\circledast:A\times A\rightarrow A$ and $a,b,c\in A$, then the expression $$a\circledast b\circledast c$$ Can be interpreted only as $(a\circledast b)\circledast c$ or is ...
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$A/B \cong A \iff B = \{e\}$ for groups $A,B$?

Let $A$ and $B$ be groups (can be infinite) Is it true that $A/B$ isomorphic to $A$ $\Leftrightarrow$ $B=\{e\}$ I didn't find a way to prove it. Thanks
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1answer
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On solutions of an equation in $\mathbb{Z}_3$

For integer numbers $x_1, x_2, y_1, y_2, y_3$ suppose that $$ x_1 + x_2 \equiv y_1 + y_2 + y_3 \quad (mod \, 3). $$ For $k=0, 1, 2$ define $$ s_k = \Big| \{ y_i \,|\, y_i \equiv k \quad (mod \, 3) ...
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1answer
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$\operatorname{End} (\Bbb Z)$ is a commutative ring while $\operatorname{End}(\Bbb{Z \times Z})$ is non commutative [on hold]

Show that $\operatorname{End} (\Bbb Z)$ is a commutative ring while $\operatorname{End}(\Bbb{Z \times Z})$ is a non commutative ring.
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1answer
66 views

Ideals and the distributive property

Does an ideal necessarily obtain the distributive property of its ring? Forgive me if the answer is obvious, I am new to ring theory. Also, any recommendation of a ring theory text would be ...
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Extension of valuation

This is a question from Milne's 'Algebraic Number Theory'. Let $K$ be a valued field with absolute value $|\cdot|$ and $L=K(\alpha)$ a finite separable extension of $K$. Let $\hat{K}$ be the ...
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Symmetric algebra

If $V$ is a vector space over the field $K$ with basis ${v_1, v_2,…,v_n}$, then the symmetric algebra $S(V)= K[v_1,v_2,..,v_n]$. The question is: If $K$ is a commutative ring, then this equality is ...
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2answers
63 views

Factor $X^7-(4+i)\in\mathbb{Q}(i)[X]$…if possible.

I think $X^7-(4+i)\in\mathbb{Q}(i)[X]$ is irreducible (simply because I don't know how to go about factoring it). Would it suffice to show that it is irreducible over $\mathbb{Z}[i]$? If so, I ...
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2answers
63 views

$V \cong V \oplus V$ as $K$ vector spaces

I am not very sure about the triviality of this problem but I can't see the solution. Problem is If $V$ is a countable dimensional vector space over field $K$, then as $K$ vector spaces $V \cong V ...