Tagged Questions

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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Let $G$ be a group of order $pq$, with $p$ and $q$ prime. Prove that the order of the center of $G$ is 1 or $pq$.

Let $G$ be a group of order $pq$, with $p$ and $q$ prime. Prove that the order of the center of $G$ is 1 or $pq$. Let me start off with what I did: Assume $G$ is abelian. Then we know ...
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vote
1answer
82 views

Length of composition series and injective homomorphisms

Let $f:M\rightarrow N$ be an injective $R$-module homomorphism. Show that $l(M)\leq l(N)$ where $l(M)$ denote the number of nonzero submodules in a composition series of $M$. My solution: Since ...
8
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1answer
431 views

An example of a commutative ring in which every primary ideal is prime

It is clear that every prime ideal in a commutative ring is primary. The converse is false; for example, in the ring $\mathbb{Z}$ the ideal $(p^2)$ is an example of a primary ideal that is not prime ...
2
votes
2answers
295 views

radical of sum of two ideals

$I$ and $J$ are ideals in $k[x_1,\cdots,x_n]$. Show that $\sqrt{I+J}=\sqrt{\sqrt{I}+\sqrt{J}}$. I have no idea how to prove it. Can someone help?
2
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1answer
50 views

Space modelled on ring

I am learning some deformation theory. The dimension of the moduli space for some deformation problem is bound by the dimension of (usually) the first cohomology group of the object that is to be ...
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1answer
170 views

$P \cap N$ is a Sylow $p$-subgroup of $N$, where $N$ is normal in $G$ and $P$ is a Sylow $p$-subgroup of $G$?

In 'A Course in Group Theory' by Humphreys, Proposition 11.14 says that if $G$ is a finite group, $P$ is a Sylow $p$-subgroup of $G$ and $N$ is a normal subgroup of $G$, then $P \cap N$ is a Sylow ...
2
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1answer
68 views

Module isomorphism, simple modules, and quotients

I'm reading R.S. Pierce's Associative Algebras. While proving a preliminary lemma to Nakayema's Lemma, the following is mentioned: Let $M$, $N$, be two $A$-modules where $N$ is a submodule of $M$ ...
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3answers
224 views

What is the meaning of the parentheses in $\phi^{-1}\left[\{\phi(g)\}\right]=gH=Hg$?

I am studying homomorphisms is groups and i saw a theorem saying: For $g$ in a group $G$, the cosets $gH$ and $Hg$ are the same, and collapsed onto the single element $\phi(g)$ by $\phi$. That is, ...
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4answers
597 views

Connections between number theory and abstract algebra.

I haven't taken abstract algebra yet, but I am curious about connections between number theory and abstract algebra. Do the proofs of things like Fermat's little theorem, the law of quadratic ...
0
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2answers
110 views

$GL(n, \mathbb{C})$ is algebraically closed? [closed]

Let $GL(n,\mathbb{C})$ the group of non-singular matrices. Is it algebraically closed? For $GL(1,\mathbb{C})$ is it true; but if I take linear combinations of elements in $GL(n,\mathbb{C})$ with ...
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3answers
170 views

Book recommendation for associative algebras

Currently, I am reading David Radford's Hopf Algebra, and I would like to pick up some representation theory of associative algebras as well since my knowledge of them is pretty shallow at the moment. ...
33
votes
1answer
458 views

A ring isomorphic to its finite polynomial rings but not to its infinite one.

I was messing with the ring $k[x_1,\dots,x_n,\dots]$ of polynomials in numerable many variables in order to solve an exercise of Atiyah, and the following question came to me and made me curious: ...
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2answers
55 views

Proof: let $f:A \to B$ with $f$ bijective, then $f{\restriction_{C} }: C \to B$ is bijective

I need the proof of following: "let $f:A \to B$ with $f$ bijective, then $f{\restriction_{C} }: C \to B$ is bijective" Thanks in advance
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1answer
96 views

a necessary and sufficient condition that homomorphic image of $R$ is a field

Let $R$ be a commutative ring with unity.Find a necessary and sufficient condition that homomorphic image of $R$ is a field I am little bit confused about the question. By a necessary and ...
8
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1answer
139 views

Why does the automorphism used to construct the group have to be non-inner?

I have a question on why a particular assumption is made that the automorphism used to construct a certain group be non-inner. In [Herstein, Topics in Algebra, p. 69], a construction of a nonabelian ...
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1answer
84 views

How does Tor and the Tensor functor interact?

So I've run into this question while doing some computations and I'm unsure if what I'm trying to show is true. Assume tensors are over $\mathbb{Z}$, is ...
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1answer
88 views

Prove that if R and S are nonzero rings then $R\times S$ is never a field.

This question has been asked on here before but I'm looking for some additional insights. The question comes from the section of the Dummit and Foote textbook on the Chinese Remainder Theorem. All of ...
3
votes
2answers
61 views

$\frac{S}{mS}$ is isomorphic to $\frac{R}{mR}$, where m is coprime to n.

I am trying to prove the following statement: Let R be a commutative ring with a unit element, and S be a subring of R of finite index n. Then $\frac{S}{mS}$ is isomorphic to $\frac{R}{mR}$, where m ...
3
votes
1answer
87 views

Monomials not in an ideal

Let $R=\mathbb{R}[x,y]$ denote the commutative ring of polynomials in two variables $x,y$ with real coefficients. Show that for each $k \in \mathbb{N}$ there exists a monomial of degree $k$ not ...
2
votes
1answer
344 views

Galois group of $x^8+2$ over $\Bbb{Q}$

This is what I did to find the Galois group for $x^8+2$: Splitting field: $$K = \Bbb{Q}(\zeta_8, \zeta_{16}2^{1/8})$$ Since $Gal(\Bbb{Q}(\zeta_8)/\Bbb{Q}) \cong Aut( \langle\zeta_8\rangle) \cong ...
2
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1answer
231 views

Calculating The Galois Group of the Splitting Field of $f=x^3-3$

If we let $f=x^3-3$ the let L be the splitting field for this polynomial I am trying to find $\Gamma(L:\mathbb{Q})$ and all intermediate field extensions. Now as this is a splitting field and finite ...
2
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1answer
119 views

Characteristic of commutative semisimple rings?

In one of my questions (Structure of the group ring of a direct product?), a statement is made for a commutative semisimple ring of characteristic $p^t, t\geq1$. Now I don't understand why there ...
2
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1answer
44 views

Definition of “invariant in a module”

What does it mean if someone say that the class of an ideal $I$ in a ring $R$ is an invariant of a module $M$?
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1answer
189 views

Generators of a cyclic group and their orders

a) Let $G = \langle a \rangle$ be a finite cyclic group. Prove that for each $b\in G$, $\langle b \rangle=G$ if and only if order of $b$ equals order of $G$. b) The previous part does not hold if $G$ ...
8
votes
1answer
240 views

How to recover the integral group ring?

I would like to solve the following exercise: Suppose $R$ is a commutative semisimple ring of characteristic $p^t, t\geq1$, and we have two finite groups $G_1=H_1 \times A_1$ and $G_2=H_2 \times ...
1
vote
1answer
60 views

Isomorphism of polynomial rings implying isomorphism of the coefficient rings [duplicate]

Let $R$ and $S$ be commutative rings. Let $x, y$ be indeterminates, and assume that one has an isomorphism $R[x] \rightarrow S[y]$ (not necessarily mapping $x$ to $y$ of course). Does this imply $R ...
3
votes
1answer
52 views

Sequences of composition factors in composition series

Suppose $\bullet$ $M$ is a left $R$-module which is both artinian and noetherian, $\bullet$ $C_1,\ldots,C_k$ is a list of the compositions factors of a composition series of $M$ listed up to ...
5
votes
2answers
106 views

Prove for $p(x) \in \mathbb R_n[x]$ there exist unique coefficients

For all $n\in\mathbb N$ define $$\mathbb R_n[x] =\{p(x) \in\mathbb R[x]: \deg(p(x)) \leq n\}.$$ Let $n\in\mathbb N$. Suppose polynomials $p_0(x),p_1(x),\dots,p_n(x)\in\mathbb R_n[x]$ have degrees ...
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0answers
47 views

Finding the value of an algebric expression

I have this expression $Ax+By+Cz$ where $x,y,z \geq 0$ and are integers. Suppose I am given a value $T$; I want to find the largest value which is less than $T$ and which cannot be generated by ...
3
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1answer
53 views

Find conditions for there to exist a morphism of rings from $\mathbb{Z}_m$ to $\mathbb{Z}_n$

I know that a necessary and sufficient condition for a ring morphism $\mathbb Z_m\to\mathbb Z_n$ to exist is that $n$ must divide $m$. However, I am having trouble understanding a proof that this ...
1
vote
2answers
92 views

How to show that switching the ring of integers for the field of rationals will yield a field?

Given: $( \{a+b \sqrt{n}\,|\,a,b\in \mathbb{Z} \},+,\times)$ and $( \{a+bi\sqrt{n}\,|\,a,b\in \mathbb{Z} \},+,\times)$ with $n\in\mathbb{N}^+$. Show that in the given example, if we switch ...
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0answers
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Show that $h \equiv 1 \pmod p$, where $h$ is the number of subgroups of order $p$ and $p$ divides the group order. [duplicate]

Let $G$ be a finite group and $p$ a prime number that divides the order of $G$. Let $h$ be the number of subgroups of $G$ of order $p$. Prove that there are $h(p-1)$ elements of order $p$ in ...
8
votes
1answer
136 views

Ring without $1$ where $\forall r\in R$, $\exists$ $n_r > 1$ such that $r^{n_r} = r$, and not all primes are maximal

On my algebra final exam, there was a problem that essentially asked the following: Let $R$ be a commutative ring such that for all $r\in R$, there exists $n_r\in\Bbb{Z}^{>1}$ with $r^{n_r} = ...
5
votes
1answer
110 views

Find degree of $\mathbb{Q}(i, \sqrt{-2})$ over $\mathbb{Q}$ and over $\mathbb{Q}(i)$

I'm studying for a qualifier and came upon this problem. Now my reasoning is that $\sqrt{-2} = i\sqrt{2}$ does not live in $\mathbb{Q}(i)$ and so ...
1
vote
1answer
47 views

$\operatorname{rank}(A\in M_{m\times n}(F)) =m \implies \exists~B\in M_{n\times m}(F)$ s.t. $AB=I_m$

Let $A ∈ M_{m×n}(F)$ be a matrix with $\operatorname{rank}(A) = m$. I just need some help showing that there exists a matrix $B ∈ M_{n×m}(F)$ such that $AB = I_m$.
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votes
0answers
94 views

field of rational functions on whole of projective n space

How to find the field of rational functions on whole of projective n space. This is done in hartstone Algebraic geometry Theorem 3.4 .chapter 1. But I am having difficulty in understanding the proof ...
1
vote
1answer
44 views

minimal polynomial of an element

I we have that $\alpha$ is the positive real root of $f=x^4-3$ then the splitting field of $f$ over $\mathbb{Q}$ is $\mathbb{Q}(\alpha,i)$ if I then want to find the degree of the extension of this ...
2
votes
2answers
67 views

Does $AB = I_m$ imply $n\geq m$?

Let $A ∈ M_{m×n}(F)$ and $B ∈ M_{n×m}(F)$ be two matrices such that $AB = I_m$. What should I be thinking to prove that $n ≥ m$?
3
votes
2answers
78 views

(sur/in)-jectivity

I'm having trouble showing this: Let $T : V → W$ be a linear map of finite dimensional vector spaces. Prove that $T$ is surjective (respectively, injective) if and only if $T^*$ is injective ...
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votes
3answers
234 views

Quotient groups of direct products: $\left(G\times H\right)/G\cong H$ and vice versa.

If $G$ and $H$ are groups. Let $G^\star= \{(a, e_H)| a\in G\}$ and $H^\star=\{(e_H, b) |b \in H\}$. Show that $(G \times H)/G^\star$ is isomorphic to $H$ and $(G \times H)/H^\star$ is ...
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1answer
75 views

A Galois Theory Question

Fix a prime $p$ and consider the equation $X^p-X-t^{-1}$ over $\mathbb F_p((t))$, the field of formal Laurent series over $\mathbb F_p$. What is the Galois group of this equation? After fumbling ...
0
votes
1answer
54 views

What is a better way to state this?

Let $T : V → W$ be a linear map of finite dimensional vector spaces. Prove that $T$ is surjective (respectively, injective) if and only if $T^*$ is injective (respectively, surjective). What is a ...
3
votes
1answer
89 views

reflection groups and hyperplane arrangement

We know that for the braid arrangement $A_\ell$ in $\mathbb{C}^\ell$: $$\Pi_{1 \leq i < j \leq \ell} (x_i - x_j)=0,$$ $\pi_1(\mathbb{C}^\ell - A_\ell) \cong PB_\ell$, where $PB_\ell$ is the pure ...
6
votes
1answer
528 views

Finding the Galois group over $\Bbb{Q}$.

If K is the splitting field of $X^8-2$ over $\Bbb{Q}$, I want to find the galois group. We know that $K=\Bbb{Q}(2^{1/8}, \zeta_8)$. So first I want to look at $Gal(\Bbb{Q}(\zeta_8)/\Bbb{Q})$ and ...
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5answers
127 views

Question about quotient groups and cosets

How to determine which representation to choose to form the quotient groups. For instance, if G = $Z$$_4$ × $Z$$_4$ and let N be the cyclic subgroup generated by (3, 2). Show that G/N is isomorphic ...
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2answers
74 views

How many different 5 cycles are there of 5 symbols

I try to calculate how many cycles there are of the form $(12345)\in S_5$. It is a little harder than I thought. My first intuition was it must be $5!$, but I'm sure that I'm overcounting. How can I ...
2
votes
2answers
91 views

Commutative Ring with Unity and Prime Characteristic

Let $D$ be a integral domain with characteristic $p>0$. It is easy to prove that $p$ is prime. Now, if $R$ is a commutative ring with unity and characteristic $p$, does $p$ be a prime number ...
2
votes
1answer
103 views

The set of zero-square elements in a commutative ring

Let $R$ be a commutative ring and let $I:=\left\{x \in R : x^2=0 \right\}$. Prove that $I$ is an ideal in $R$ or give a counterexample. Remark: This is problem 3B in the January 2003 Algebra ...
0
votes
1answer
52 views

Finitely generated module as a quotient of a free module of finite rank

If an $R$-module $M$ is generated by $n$ elements then how to show that $M$ can be realised as a quotient of $R^n$?
2
votes
1answer
116 views

Annihilator of an element of a left module

How to show that the annihilator of an element of a left module is a left ideal but not necessarily a two-sided ideal. When does this become an ideal?