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Questions tagged [abstract-algebra]

For questions about groups, rings, fields, vector spaces, modules and other algebraic objects. Associate with related tags like group-theory, ring-theory, modules, etc. to clarify which topic of abstract algebra is most related to your question and help other users when searching.

0
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0answers
12 views

What are the patterns for these decompositions of permutations into transpositions?

By looking at answers on this site, in order to decompose a permutation into transpositions, so far I have seen 3 patterns as follows: (12345)=(15)(14)(13)(12) (12345)=(12)(23)(34)(45) (12)=(21) ...
4
votes
2answers
141 views
+50

What is the dicyclic group of order $12$? (What is $\mathbb{Z}_3\rtimes \mathbb{Z}_4$)

I have come across the dicyclic group of order $12$. I can see that this is generated by three elements subject to some relations. Is there a way to realize this group without talking about generators ...
1
vote
1answer
53 views

Possibilities for a certain subgroup of $S_n$.

A two-part problem: $\bullet$ Let $H \leq S_n$ be the subgroup that fixes 1. Show that $H$ is isomorphic to $S_{n-1}$. $\bullet$ Show that there are no proper subgroups of $S_n$ that properly ...
-2
votes
2answers
29 views

If $a \in \mathbb{C}$ and $\exists n \in \mathbb{N}$ s.t. $\{ a^n, a^{n+1} \} \in \mathbb{N}$, prove $a \in \mathbb{N}$ [on hold]

Let $a$ be a complex number. If it exists a natural number $n$ (different of $0$), such that $a^n$ and $a^{n+1}$ are integers, prove that $a$ is an integer.
2
votes
1answer
49 views

Defining the most natural map between two modules

Let $R = \mathbb{C}[t]$ be a ring of polynomials in variable $t$ with coefficients in the field of complex numbers $\mathbb{C}$ and let $$M = R[x]/(tx-1).$$ Goal: I need to show that $$M \cong R[t^{-...
2
votes
2answers
121 views

Prove that $\det(\varphi(\sigma)) = \operatorname{sgn}(\sigma)$ for all $\sigma \in S_n$.

Fix a positive integer $n$. For each $\sigma \in S_n$ define an $n\times n$ matrix, $\varphi(σ)$ by $$\varphi (\sigma)_{i,j}= \begin{cases} 1 & \text{if } i = \sigma (j), \\ 0 & \...
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0answers
8 views

Localization and completion

Eisenbud, Exercise 7.1: Let $m$ be a maximal ideal of a ring $R$. Show that the map $R\to \hat{R}_m$ factors through the localization map $R\to R_m$. Here $\hat R_m$ is the completion of $R$ w.r.t....
3
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1answer
24 views

Does transitivity of a relation imply associativity of an operation?

Let $*$ be a binary operation on a set $S$. We define a binary relation $R$ on $S$ by: $xRy$ iff $\exists z: x*z=y$. If $*$ is associative, then the relation $R$ is transitive. My question is whether ...
2
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1answer
35 views

Why is m annihilating M/mM?

Let R be a Noetherian local ring with maximal ideal m. If M is an R-module, why is m the annihilator of M/mM?
1
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1answer
24 views

some questions on finding the Galois group of the splitting field of $x^4-3$

I have two question regarding the Galois group of the splitting field of $x^4-3$. Firstly I know the roots of this polynomial are $\sqrt[4]{3},w\sqrt[4]{3},w^2\sqrt[4]{3},w^3\sqrt[4]{3}$, where $w$ ...
0
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1answer
14 views

Prove that ((𝑁∩𝐻)𝑀)/𝑀 is a subgroup of 𝑁/𝑀. Then, Deduce ((𝑁∩𝐻)𝑀)/𝑀 is Abelian.

Suppose that 𝐻,𝑁,𝑀 are subgroups of 𝐺, 𝑀 is a normal subgroup of 𝑁. Assume 𝑁/𝑀 is Abelian. Prove that ((𝑁∩𝐻)𝑀)/𝑀 is a subgroup of 𝑁/𝑀. Then, Deduce ((𝑁∩𝐻)𝑀)/𝑀 is Abelian. I can ...
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0answers
13 views

A question about the definition of path algebras

A quiver $Q=(Q_0,Q_1, s,t)$ is a quadruple consisting of a set $Q_0$ of vertices, a set $Q_1$ of arrows, and two maps $s,t :Q_1\rightarrow Q_0$ which associate each arrow $a\in Q_1$ to its source ...
4
votes
0answers
42 views

Prove that $[\mathbb{Q}(\sqrt[4]{10}\zeta_8,i):\mathbb{Q}]=8$

Determine the Galois group of $f=X^4+10$ and the lattice of intermediary fields. I know that $f$ is irreducible by Eisenstein for $p=5$. I calculated the roots to be $\sqrt[4]{10}\zeta_8^i$ for $i\in\...
0
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1answer
18 views

$F(f)=0$, then $f=0$ where $0$ is the zero morphism for equivalent functors.

For an additive category $\mathcal{C}$ and let $F:\mathcal{C} \to \mathcal{C}$ be a equivalent additive and covariant functor. Prove that for every morphsim $f:A \to B$ such $F(f)=0$, then $f=0$ ...
0
votes
1answer
18 views

Suppose that $H,N,M$ are subgroups of $G$, $M$ is a normal subgroup of $N$. Show that $(N\cap H)M$ is a subgroup of $N$.

Suppose that $H,N,M$ are subgroups of $G$, $M$ is a normal subgroup of $N$. Show that $(N\cap H)M$ is a subgroup of $N$. I know all elements in $(N\cap H)M$ should be also in $N$. I just need to test ...
2
votes
2answers
47 views

Non-Separable Polynomials and their Derivatives

We say that a polynomial $f(x) ∈ F[x]$ is inseparable if it has a repeated root in some field extension. Otherwise we say that $f(x)$ is separable. Prove that $f(x)$ is separable $\iff\gcd(f, Df) = 1$....
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0answers
53 views

Why is $I\subset J$ necessary for defining the quotient $J/I$?

Let $J$ be a proper ideal of a Ring $R$.Then $J/I$ is a proper ideal of $R/I$ if $I$ is contained in $J$ and $I$ is an ideal of $J$. My question is why the condition "$I$ is contained in $J$" ...
0
votes
1answer
38 views

Formal definition of “ inverse operations” on a set?

I'd like to express formally the fact that multiplication and division are inverse operations. In order to do that, I'd like to find a general definition of " inverse operations" on a set. Suppose ...
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2answers
53 views
+50

Can Rings be viewed as “function rings over their spectrum”?

In the nlab article about localization, a side note says When interpreting a ring under Isbell duality as the ring of functions on some space $X$ (its spectrum), […] Unfortunately, the article ...
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2answers
16 views

$H$ is a subgroup of $G$ with $\phi(H) = H$ $\forall \phi \in Aut(G) \Rightarrow $ $H$ is normal in $G$ [duplicate]

$H$ is a subgroup of $G$ with $\phi(H) = H$ $\forall \phi \in Aut(G) \Rightarrow $ $H$ is normal in $G$ I'm not sure where to begin. Perhaps we can use this corollary of the First Isomorphism ...
1
vote
1answer
33 views

Direct sum of $n$ (infinite) cyclic groups isomorphic to direct sum of $n$ copies of $\mathbb{Z}$?

I'm currently selfstudying some algebra and i am currently covering the various equivalent definitions of free abelian groups. However, in order to understand why these definitions are indeed ...
-1
votes
3answers
76 views

Let $T:V → V$ be a linear transformation where $V$ is finite dimensional. [closed]

Let $T:V → V$ be a linear transformation where $V$ is finite dimensional. Show that exactly one of (i) and (ii) holds: (i) $T(v)=0$ for some $v ≠ 0$ in $V$ ; (ii) $T(x)=v$ has a solution $x$ in $V$ ...
1
vote
0answers
17 views

Bar resolution and the morphisms define on $B_n$(free module)

Given a group extension $0\to K\to G\to Q\to 1$, we define $B_n$ as the free $\Bbb Z[Q]$-module on $Q^n$. And then we want to make a exact sequence $\cdots\to B_3\to B_2\to B_1\to B_0$, where the ...
2
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0answers
730 views

Proof of every finite group is finitely presented.

I'm reading the proof that every finite group is finitely presented from Dummit's Abstract Algebra, but there's a part that I don't understand. In the proof below, what are the elements $\tilde{g_i}$? ...
2
votes
1answer
1k views

Class Equation : 20 = 1 + 4 + 5 + 5 + 5, normal subgroup of order 5 not normal subgroup of order 4

I have the following class equation: 20 = 1 + 4 + 5 + 5 + 5 I know that there are subgroups of order 4 and 5 in G. I see this because |G| = |centralizer| * |conjugacy class| and the centralizer is a ...
4
votes
3answers
189 views

$T:V\rightarrow W$ such that $N(T)=V'\subset V$ and $R(T)=W'\subset W$

How should one prove that there exists a linear map $T:V\rightarrow W$ such that $N(T)=V'\subset V$ and $R(T)=W'\subset W$ if $\dim(V')+\dim(W')=\dim(V)$, where $V$ and $W$ are finite-dimensional ...
0
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0answers
20 views

Group theory : representation Matrix of generating element for families of functions?

Assume I have some sampling of a function $f(t)$ at points $t$: $$f(t_k) = d_k, \forall k \in \{1,\cdots,n\}$$ Assume we have vectors $\bf v_k$ which can be functions of $x_k$ and $d_k$, can we find ...
0
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0answers
11 views

mapping matrix for a simple model

consider X=[x11 x12 x13 x21 x22 x23] as 2*3 matrix and U=[u1 u2 u3 u4 u5]' as 5*1 vector (column vector) so that x ...
0
votes
0answers
27 views

Why $R[G]/\ker\varepsilon\cong V_0(R)$, where $R[G]$ is a group ring and $V_0(R)$ a trivial $R[G]$-module?

Given a group $G$ and a commutative ring $R$. Then we can produce a group ring $R[G]$, and there exists a map $\varepsilon:R[G]\to R$. Moreover, $R[G]/\ker\varepsilon\cong V_0(R)$, where $V_0(R)$ is a ...
1
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3answers
29 views

Equivalence classes – Topics in Algebra Herstein.

Property $2$ of an equivalence relation states that if $a \sim b$ then $b \sim a$; property $3$ states that if a $a \sim b$ and $b \sim c$ then $a \sim c$. What is wrong with the following proof ...
0
votes
1answer
50 views

Prove $G=N_G(P)H$ where $H \lhd G$ and $P \in Syl_p(H)$. [duplicate]

Prove $G=N_G(P)H$ where $H \lhd G$ and $P \in Syl_p(H)$. I definitely need some help with this one. I was thinking that maybe the problem meant to say that $P$ was a sylow subgroup of $G$, that would ...
0
votes
2answers
56 views

How many group homomorphisms are there from the alternating group $A_5$ to symmetric group $S_4$? [on hold]

How many group homomorphisms are there from the alternating group $A_5$ to symmetric group $S_4$? Is there any shortcut to find?
0
votes
1answer
21 views

Finding an irreducible polynomial in $\mathbb{Z}[x]$ such that it is reducible modulo 2,3 and 5. [duplicate]

The problem is finding an irreducible polynomial in $\mathbb{Z}[x]$ such that it is reducible modulo 2,3 and 5. I can't find anything, any help is appreciated. (Is there some general strategy for ...
1
vote
1answer
24 views

Tensor product of Galois extension

Let $K/k$ be a finite Galois extension of fields with Galois group $G$. How to show that the (n+1)-fold tensor product $$K \otimes_k K \otimes_k K \cdots \otimes_k K$$ is isomorphic to $$\prod\...
0
votes
1answer
20 views

Product of elements from Direct summand in the direct sum abelian group

Let $G = H \oplus K$ be abelian group. Now, I follow the definition of direct sum from Wikipedia which is Now, we choose two element (both of them are not identity) $h \in H$ and $k \in K$ ($h, k \...
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0answers
24 views

“Neutral element” of a group action in the set [duplicate]

I have difficulties solving the following problem, I would be grateful for any tipps or advice: a) Let $p$ be a prime and $G$ a group with $|G|=p^n$. $G$ operates on an finite set $X$ with $p \nmid |...
0
votes
1answer
24 views

$f$ is an automorphism and $f(5)=5$ then what are the possibilities for $f(1)$

Suppose $f$ is a map from $\mathbb{Z}_{20}\mapsto \mathbb{Z}_{20}$. Let us assume that $f$ is an group automorphism and $f(5)=5$ then what are the possibilities for $f(1)$? Now first thing I know is ...
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0answers
51 views
+50

Prime ideals of $R[x]$ that intersect $R$ in $P$

Let $R$ be a noetherian ring and $P$ a prime ideal of height $h$. Show that the prime ideals $Q\subset R[x]$ that intersect $R$ in $P$ are of the following two kinds, with height as shown: $Q=...
3
votes
0answers
37 views

Hungerford Chapter 2 Section 2 Problem 2 WITHOUT using the structure theorem of finite abelian groups

Let $G$ be a finite abelian group and $x$ an element of maximal order. Show that$\langle x \rangle$is a direct summand of $G$. Use this to obtain another proof of Theorem 2.1. Theorem 2.1: Every ...
1
vote
1answer
622 views

The number of automorphisms of a finite field

Let $M$ be a finite field and $|M| = p^s$, where $p$ is prime and $s \in \mathbb N$. Prove that the number of different isomorphisms field $M$ to $M$ equal to $s$ and this isomorphisms form a cyclic ...
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0answers
30 views

When degree of splitting field equals n factorial

Given that the degree of splitting field of a polynomial $f(x)$ over $\mathbb{Q}[x]$ is equal to $n!$ where $n$ is the degree of $f$, $n>2$. If $\alpha$ is a root of $f$ in the splitting field, ...
2
votes
1answer
70 views

Dimension about space of matrices of order 3 over the field of the real.

Consider the vector space of the matrices of order 3 over the field of the real $M_{3}\left(\Re\right)$ numbers. and let S be the subspace such that is spanned by the matrices of the form $AB-BA$. ...
1
vote
1answer
100 views

Pythagorean closure

Im reading the book "Galois Theory" by Ian Stewart $(4$th Edition$)$. Here the author defines the Pythagorean closure as follows: Definition:The Pythagorean closure $\mathbb{Q}^{PY}$ of $\mathbb{Q}...
3
votes
1answer
62 views

Galois Theory and AR Theorem

I’m trying to understand the proof that there will never be a quintic formula. Some questions: What is the name for a number which is formed from progressive radicals and algebraic operations, i.e. a ...
0
votes
1answer
24 views

In a Euclidean ring $R$, prove $(a) ⊆ (b) \iff b|a$

Let $a, b$ be elements of a Euclidean ring $R$. Prove that $$(a) \subseteq (b) \iff b \;\text{divides}\;a.$$ I have no clue how to even start this. Any help would be great, thank you in advance!
1
vote
3answers
56 views

Given a group $H\unlhd G$, is it natural/accurate to say that $G \cong H \oplus (G / H)?$ [duplicate]

Given a group $G$ and a normal subgroup $H \leq G$ , is it natural / accurate to say that $$G \cong H \oplus (G / H)$$ This is what I feel is true intuitively, but I'm sure a group theorist could ...
0
votes
0answers
33 views

Let $R$ be a ring such that $\Bbb Z \subset R \subset \Bbb Q$. Show that $R$ is a PID. [duplicate]

Problem: Let $R$ be a ring such that $\Bbb Z \subset R \subset \Bbb Q$. Show that $R$ is a PID. Hint: If $I$ is an ideal of $R$ consider $A=\Bbb Z \cap I$. Following the hint, if $I$ is an ideal of $...
0
votes
1answer
29 views

Let 𝐿/𝑄 be a field extension. Show that complex conjugation defines an element of Aut𝑄(𝐿).

Let 𝐿/𝑄 be a field extension. Show that complex conjugation defines an element of Aut𝑄(𝐿). I am not sure about how to show it. Isn't it intuitive from the fundamental theorem of Galois theory?
0
votes
1answer
27 views

Proof of principle ideal

Let $I = \{p(x) ∈ \Bbb Z[x]: 5\mid p(0)\}$. Prove that $I$ is an ideal of $\Bbb Z[x]$ by finding a ring morphism from $\Bbb Z[x]$ to $\Bbb Z_5$ with kernel $I$. Prove that $I$ is not a principal ideal....
6
votes
0answers
51 views

$R^{\mathbb N}$ as a free $R$-module.

Suppose that $R$ is a commutative ring. I'm wondering if the space $R^{\mathbb N}$ is a free $R$ module. I know how to prove that it is not a free $R$ module in the case of $R = \mathbb Z$. But the ...