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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Generators of the group of invertible elements of the ring $\mathbb{Z}_{14}$—are they multiplicative or additive?

When I was asked to find the generators of the group of invertible elements of the ring $\mathbb{Z}_{14}$, which are denoted as $\phi(14)$, I did not realize whether the generators are multiplicative ...
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1answer
33 views

Why is the commutator group a subgroup?

I am in Intro to Algebra, and have a question regarding the commutator subgroup. I am a bit confused with the premise, though, with how the set is a subgroup in the first place. Let $C$ be the set of ...
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1answer
15 views

Number of equivalence Relations containing $(1,2)$

Find the number of equivalence Relations on the Set $A=\{1,2,3 \}$ which contains the Element $(1,2)$. My Try: Since $(1,2)$ is to be included, so is $(2,1)$ since the Relation should be Symmetric ...
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28 views

How to compute $Z_n \times Z^*_m$

How to compute $Z_n \times Z^*_m$.(In journal Multiplicative Properties of Set Residues) say by chinese remainder theorem, may be thought of as $\left\{a \in Z_{mn}:(a,m)=1\right\}$. where $Z^*m$ unit ...
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3answers
50 views

Looking for help to understand example of Group

I am looking for someone to help me to understand what is going on in the following example, from Hersteins "Topics in Algebra". It says, Let $G$ be the set of all $2*2$ matrices $$\begin{pmatrix} ...
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1answer
20 views

Unique factorization domain: $\mathbb{Z}_{n}[x]$

How to determine all $n\in\mathbb{N}$ such that $\mathbb{Z}_{n}[x]$ is a unique factorization domain? I am guessing that this would be true for all primes, since $\mathbb{Z}_n$ is a UFD when $n$ is ...
5
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1answer
30 views

$F[[T]] \times F[[1/T]]$, fundamental domain.

Let $p$ be a prime number. Here is a link which shows how to see that $$(\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T)))/\mathbb{F}_p[T, 1/T]$$is compact using an adelic result. (Here $\mathbb{F}_p[T, ...
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0answers
96 views

Subring of $\mathcal O(\mathbb C)$

Let $\mathfrak A \subset \mathcal O(\mathbb C)$ be the subring generated by the nowhere zero analytic functions $f: \mathbb C \to \mathbb C$. Does we have a precise description of $\mathfrak A$ ? Is ...
6
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39 views

$y^2 = x^3 - 26$, exist ideal satisfying conditions?

For the solution $(x, y) = (3, 1)$ of $y^2 = x^3 - 26$, does there necessarily exist an ideal $I$ of the integer ring $\mathbb{Z}[\sqrt{-26}]$ of $\mathbb{Q}(\sqrt{-26})$ such that $(y + \sqrt{-26}) = ...
3
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1answer
27 views

If $G$ is a group of order $2^nm$, where $m$ is odd and $(m-1)!<2^n$, show that $G$ is not simple.

If $G$ is a group of order $2^nm$, where $m$ is odd and $(m-1)!<2^n$, show that $G$ is not simple. I started out by trying to prove this using the Sylow theorem, but it led nowhere. I was able to ...
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1answer
54 views

Ideal in $\mathcal O(\mathbb C)$

Let $\mathfrak {I}$ the ideal generated by all the holomorphic functions which are never zero. Question : is $\mathfrak {I} = \mathcal O(\mathbb C)$ ?
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2answers
32 views

From group isomorphisms to algebra isomorphisms

Let $A$ be an algebra and let $A^{\ast}$ be the subset of units (that is, invertible elements) of $A$. Then $A^{\ast}$ is a group under the multiplication of $A$. Let $f^{\ast}:A^{\ast}\to A^{\ast}$ ...
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0answers
12 views

every $F$-algebraic homomorphism of $K$ is $1-1$ and onto? [duplicate]

Suppose that $K|F$ is a field extension of finite degree. We know that every $F$-algebraic homomorphism of $K$ is $1-1$ and onto. Also we know that every finite field extension is algebraic extension. ...
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0answers
25 views

Is it true that $M\otimes_A F\simeq M^{(I)}$?

Let $A$ be an $R$-algebra ($R$ is a commutative ring with identity $1_R$) and suppose $F$ is a left free module over $A$. Is it true that $$M\otimes_A F\simeq M^{(I)}$$ for any right module $M$ over ...
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1answer
35 views

Acting algebraically

Let $G$ be a group that acts on some non empty set $V$. What does it mean that $G$ acts algebraically on $V$? I am well aware of the definition of being algebraic. But I cant find the definition ...
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2answers
31 views

Which of the following about a permutation is correct?? (CSIR-2015, June)

Let $\sigma:\{1,2,3,4,5\}\rightarrow\{1,2,3,4,5\}$ be a permutation (one-to-one and onto function) such that $$ \sigma^{-1}(j)\le \sigma(j) \quad\text{for all $j$ such that }1\le j\le 5. $$ Then ...
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13 views

Normal Submagma?

Is there a definition of normal submagma? visit https://en.wikipedia.org/wiki/Magma_(algebra) For normal sub-quasi-group I found two: A sub-quasi-group $H$ is called normal if there exists a normal ...
2
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0answers
35 views

Application of the structure theorem for finitely generated modules over a PID

I've been trying to figure out the correct way to do this and I'm not sure I've got it quite right. $\mathbb{Z}^4/S$ where $S = \{(4, 12,20,8),(6,30,18,12),(4,16,16,8),(8,40,24,16) \}$ So, $M_0 = ...
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1answer
14 views

Explicitly compute the trace for the tautological representation of $D_4$ of $\mathbb{R}^2$.

Fix a finite dimensional representation $\rho: G \longrightarrow GL(V)$ of $G$. Its trace is defined as the function $tr:G \longrightarrow F$ defined by $tr(g) = tr(\rho(g))$. Explicitly compute ...
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2answers
76 views

Hardcore Abstract Algebra Book Request

I am going to start learning Abstract Algebra soon. I was originally going to start with Dummit and Foote, but I am starting to abandon that idea. I want to use a "hardcore" algebra book. I don't mind ...
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1answer
29 views

Non-zero maps between modules

Let $R$ be a ring and $M,N$ two $R$-modules such that there exists a non-zero map $\psi : M \to N$. Is it true that there exists a non-zero map $\varphi: N \to M$ ? I do not have any particular ...
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36 views

The ordinals as a free monoid-like entity

Let $\kappa$ denote a fixed but arbitrary inaccessible cardinal. Call a set $\kappa$-small iff its cardinality is strictly less than $\kappa$. By a $\kappa$-suplattice, I mean a partially ordered set ...
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1answer
43 views

Follow up question to finding primes $p$ such that $f(x)=x^6 - x^3 +1$ factors (in various ways) in $\mathbb{F}_p$

I asked this question yesterday, however, I am not sure how to compare the solution given on this site to the "worked example" solution as in my notes. $\textbf{Problem Statement:} $ Let $f(x)= x^6 - ...
5
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2answers
68 views

$L$-function, easiest way to see the following sum?

What is the easiest way to see that$$\sum_{(m, n) \in \mathbb{Z}^2 \setminus \{0, 0\}} (m^2 + n^2)^{-s} = 4\zeta(s)L(s, \chi)?$$Here $\chi$ is the homomorphism $(\mathbb{Z}/4\mathbb{Z})^\times \to ...
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0answers
24 views

Galois group isomorphic to $\mathfrak{S}_5$.

Let $f(x) \in \mathbb{Q}[x]$ an irreducible polynomial of degree $5$ and with splitting field $K \supset \mathbb{Q}$. If $\mathbb{Q}(\sqrt{7})$ and $\mathbb{Q}(\sqrt{11})$ are subfields of $K$,is ...
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0answers
37 views

Group isomorphism exercise

I am trying to solve the following problem: Let $G$ be a finite group such that there exists an isomorphism $\phi:f:G/Z(G) \to \mathbb Z_3 \oplus \mathbb Z_3$ and $x \in G$ such that ...
4
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4answers
75 views

what is the greatest integer that divides $p^4-1$ for every prime number p greater than 5

what is the greatest integer that divides $p^4-1$ for every prime number p greater than 5(this is a gre subject math problem) I think that $p^4-1=(p^2+1)(p-1)(p+1)$,so 8 must divide all the $p^4-1$ ...
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1answer
37 views

Subgroup proof verification.

Let $G$ be an abelian group, K is a fixed positive integer. $H$={$a\in$ $G$ $|$ $|a|$ divides K} . Prove that $H$ is a subgroup of $G$. My way of proving (Let me know how I could make it better or ...
3
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1answer
55 views

Exercise from Serre's “Trees” - prove that a given group is trivial

In Serre's book "Trees" on page 10 the following exercise is given: Show that the group defined by the presentation $$x_2x_1x_2^{-1}=x_1^2, \hspace{7pt} x_3x_2x_3^{-1}=x_2^2, \hspace{7pt} ...
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0answers
38 views

Let I be an ideal of a ring R. Prove that the quotient ring R/I is a commutative ring if and only if ab − ba ∈ I for all a, b ∈ R. [on hold]

This is the report no. 3 of Jennylou Canlas in our subject math126 in MSU Proof: Suppose R/I is a commutative ring. Let a, b ∈ R. Then (a + I), (b + I) ∈ R/I. Since R/I is commutative , (a + I)(b ...
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2answers
24 views

Let I and J be ideals of a ring R. Prove that the union of I and J is an ideal of R iff I is a subset of J or J is a subset of I. [duplicate]

I'd like a proof of: Let I and J be ideals of a ring R. Prove that the union of I and J is an ideal of R iff I is a subset of J or J is a subset of I. This is my report no. 3 in my subject ...
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1answer
55 views

Are $\mathbb{N}$ is isomorphic to $\mathbb{Q}$? [duplicate]

Are $\mathbb{N}$ isomorphic to $\mathbb{Q}$? There are any difference between isomorphism and cardinal equality? If $X$ and $Y$ are two sets and $\text{Cardinal}(X)=\text{Cardinal}(Y)$, is $X$ ...
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1answer
36 views

Question about the quotient map of commutative algebra [on hold]

Let $A$ be a commutative algebra and let $I$ be an ideal of $A$. Denote the quotient map from $A$ to $A/I$ by $f$. Let $J$ be a maximal ideal of $A/I$. Is it true that $f^{-1}(J)$ is a maximal ideal ...
6
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1answer
43 views

Quadratic field, $O_K/\mathfrak{p} = \mathbb{F}_p$, $O_K/pO_K$ is a finite field of order $p^2$.

Let $K$ be a quadratic field $\mathbb{Q}(\sqrt{m})$ where $m$ is a square free integer, and let $p$ be a prime number which does not divide $2m$. Where can I find a reference to a proof of the ...
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1answer
79 views

Group as a $\mathbb Q$-vector space

Let $G$ be a torsion free abelian group of having $n$ number of maximally rationally independent elements $r_{1}, r_{2}, ..., r_{n}$ and assume that $G$ is not finitely generated. Is this correct ...
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1answer
32 views

Prove that there's a unique morphism that completes the commutative diagram

I have to prove that there's a unique $\gamma : M'' \rightarrow N''$ that completes this diagram considering the rows are exact. $$\begin{array} MM' \stackrel{f_1}{\longrightarrow} & M & ...
0
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1answer
54 views

Prove that $\mathbb{Q}{[\sqrt3]} = \lbrace a + b\sqrt3: a, b \in \mathbb{Q} \rbrace$ is a subfield of $\mathbb{R}$

Prove that $\mathbb{Q}{[\sqrt3]} = \lbrace a + b\sqrt3: a, b \in \mathbb{Q} \rbrace$ is a subfield of $\mathbb{R}$ My answer: Must show: i) $0 \in \mathbb{Q}{[\sqrt3]}$ ii) $1 \in ...
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0answers
26 views

Are there infinite-dimensional, artinian C*-algebras?

A ring is artinian if it has no infinite descending chains of ideals. Of course finite-dimensional algebras are artinian. I'm wondering if it's possible to have an artinian C*-algebra (or Banach ...
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3answers
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Find a prime $p$ such that $f(x)=x^6 - x^3 +1$ factors in to linear factors in $\mathbb{F}_p[x]$

Find a prime $p$ such that $f(x)=x^6 - x^3 +1$ factors in to linear factors in $\mathbb{F}_p[x]$ $\textbf{My attempt:}$ Notice that $f(x)$ is the $18$-th cyclotomic polynomial, $\Phi_{18}(x)$. For ...
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1answer
42 views

Group of order $2m$ where $m$ odd has a subgroup of index 2. [duplicate]

Show that a group $G$ of order $2m$, where $m$ is odd, has a subgroup of index $2$. I am feeling a little dubious about my proof. Let $G$ act on itself by left multiplication to induce the ...
2
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1answer
30 views

Another approach to $A/I\otimes_A A/J\simeq A/(I+J)$?

The same question appears here $A/ I \otimes_A A/J \cong A/(I+J)$ however I'm looking for a different approach. Let $A$ be an algebra, $I$ a right ideal and $J$ a left ideal. I'd like to show ...
2
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1answer
51 views

Looking for various proofs there is no faithful representation $\rho:S_4\rightarrow GL(\mathbb{R}^2)$

I stumbled on the question when thinking about a representation $\rho:D_4\rightarrow GL(\mathbb{R}^2)$ as symmetries of the four points $\{(\pm 1, \pm 1)\}$. There didn't seem to be a good geometric ...
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3answers
67 views

Find all of the homomorphisms $ \varphi: S_3 \to \mathbb{Z}_{4} $.

I'd like to find all of the homomorphisms $ \varphi: S_3 \to \mathbb{Z}_{4} $. What I've tried so far: I tried to do $ \varphi(Id) = \bar{0} = \bar{4} $ (as somebody used here). But then I realized ...
2
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2answers
44 views

Finitely generated abelian group with certain properties

Problem Characterize all finitely generated abelian $G$ such that every proper subgroup of $G$ is cyclic, $G$ contains exactly two proper subgroups, and for each pair of subgroups $S$,$T$ in $G$ ...
4
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2answers
46 views

Groups of the from $gMg$ in a monoid where $g$ is an idempotent

Let $(M, \cdot)$ be a finite monoid with identity $e$. It is easy to see that $gMg = \{ gxg : x \in M \}$ forms a monoid with identity $geg = g$ if $g$ is an idempotent. If $gMg$ contains no ...
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1answer
59 views

Infinitude of the primes $p\equiv1 \operatorname{mod} n$

$\textbf{Theorem:}$ Fix $1 < n \in \mathbb{Z}$. There are infinitely many primes $p\equiv1 \operatorname{mod} n$. $\textbf{Proof}$ Recall that the $n$-th cyclotomic polynomial $\Phi_n(x)$ is ...
7
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1answer
86 views

What is the Galois group of $\mathbb{Q}_p[\zeta] / \mathbb{Q}_p$, where $\zeta$ is a $p^r$th root of unity?

Let $\zeta$ be a $p^r$-th root of unity and $\mathbb{Q}_p$ the p-adic numbers. I would like to know the Galois automorphisms of $\mathbb{Q}_p[\zeta] / \mathbb{Q}_p$. I already know, the degree of ...
0
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1answer
33 views

Ring Structure for Non Commutative Groups: Is there a grander reason for Abelian requirements?

So I was considering the following idea. Let a generalized Ring $gR$ be a set equipped with two operations $$u_1, u_2$$ Such that $gR$ is a with the operation $u_1$ and the operation $u_2$ ...
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1answer
12 views

dimension of the quotient of a bialgebra

I am stuck in a proof of a lemma that I am in need of. The situation is as follows: Let $k$ be a field and $A$ and $B$ two finite-dimensional $k$-bialgebras, where the dimension of $A$ is a prime ...
3
votes
2answers
46 views

How to choose a left-add$(X)$-approximation with a certain property

Let $A$ be an artin algebra and $X,Y$ in mod-$A$. Suppose $0\rightarrow Y \stackrel{\alpha}{\rightarrow} X^n\stackrel{\beta}{\rightarrow} X^m$ is exact. Set $C:=Coker(\alpha)$ (as module) and ...