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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3
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Centralizer of $\mathbb{C}[G]$ in $\mathbb{C}[H]$

I found this result, but can't understand how to prove. Let $H$ be a subgroup of $G$. Then prove $Z(\mathbb{C}[G],\mathbb{C}[H])$ is commutative iff every irreducible $G$ module when restricted to ...
4
votes
1answer
27 views

Representing Matrix Subgroups

Suppose I wanted to describe the subgroup of $GL_n(\mathbb{R})$ of matrices of the form $$ \left [ \begin{array}{cc} A & B \\ 0 & C \\ \end{array} \right ] $$ where $A \in ...
11
votes
4answers
372 views

Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$

I can't crack this one. Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$ (the only constraint is that $a,b,c,z \in \mathbb{Z}$).
1
vote
2answers
28 views

For $\mathbb{Q[x]}/ I \cong$ $\mathbb{Q}$, proving kernel

Let I $=<x-2>$. Prove $\mathbb{Q[x]}/I \cong \mathbb{Q}$ $\textbf{Pf:}$ Define $\phi: \mathbb{Q[x]} \rightarrow \mathbb{Q}$ by $\phi(f(x)) = f(2)$ I understand how to show that it is ...
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0answers
24 views

When is the normality of field extensions transitive?

I want to answer the following question: Given field extensions $K \subset M \subset L $, when is the case that $M:K$ is normal and $L:M$ is normal implies $L:K$ is normal? In my algebra class we have ...
1
vote
1answer
15 views

Galois groups of extensions over $\mathbb{Q}$

Let $K=\mathbb{Q}(\sqrt[8]{2},i)$ and let $F=\mathbb{Q}(i)$. I need to show that $\operatorname{Gal}(K/F)\cong Z_8$ where $Z_8$ is the cyclic group of order 8. I have already shown that $[K:F]=8$, ...
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2answers
30 views

Classifying groups of order 4

Let $G$ be a group of order $4$ then either $ G \cong C_4$ or $G \cong C_2 \times C_2$ the proof my lecture gave goes as follows: Let $x \in G$ then by Lagrange $\text{ord}(x)$ divides $|G| =4 $ so ...
3
votes
1answer
67 views

What does the notation $\mathbf{R}^\mathbf{R}$ mean?

I was reading the Princeton Review of GRE math subject test (4th edition), and one question was (page. 251) Example 6.24 Is the ring $\mathbf{R}^\mathbf{R}$ an integral domain? ...
2
votes
1answer
51 views

When is $\mathbb{Q}(x)$ a finite extension of $\mathbb{Q}$?

Let $x\in\mathbb{C}$, and consider the field $\mathbb{Q}(x)$. If this field is a finite extension of $\mathbb{Q}$, then $x$ is algebraic over $\mathbb{Q}$, so satisfies some polynomial $P$. Any field ...
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2answers
13 views

the cardinal of the set of left cosets the same as the cardinal of the set of right cosets?

is $|G/H| = |H/G|$ where $G/H$ is the set of left cosets of H in G, and $H/G$ the set of right cosets of H in G? I know that $|gH| = |H| = |Hg|$ but I don't see how $|G/H| = |H/G|$, even though ...
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4answers
64 views

Ring homomorphism from $M_3(\mathbb{R})$ into $\mathbb{R}$

I was working on this problem Find all ring homomorphisms from $M_3(\mathbb{R})$ into $\mathbb{R}$. My attempt:- I found that if we have any ring homomorphism $\phi$, then $\ker(\phi)$ should ...
0
votes
0answers
31 views

What do we know about a group when we know it's order? [on hold]

Specifically, if a Group has order $rs$, what can we automatically know about this group?
0
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1answer
24 views

Defining an operation on a quotient set

I found the following exercise in the beginning of Ames An Introduction to Abstract Algebra. (I left out a few parts.) $f$ is a map $G\mapsto G'$ of two groups. For any $x'\in Gf,$ take ...
3
votes
1answer
44 views

Length of tensor product of finite length modules is finite

Let $R$ be a commutative ring. If $M$ and $N$ are finite length $R$-modules, then $M\otimes_R N$ has finite length, and $l(M\otimes_R N) \le l(M)l(N)$. I know the question has been posted ...
-1
votes
3answers
71 views

find the minimal polynomial over $\mathbb{Q}$ of the algebraic number $\mathbb{(1+\sqrt{5})/2}$ [on hold]

find the minimal polynomial over $\mathbb{Q}$ of the algebraic number $\mathbb{(1+\sqrt{5})/2}$ . and write down the conjugates of the number over $\mathbb{Q}$
2
votes
0answers
24 views

Examples for Burnside problem.

What are some examples for Burnside Problem- example of an infinite finitely generated torsion group - except Grigorchuk group. I have studies Grigorchuk group as an counterexample which was first ...
-2
votes
1answer
57 views

Using an isomorphism to define “sameness” for groups [closed]

Having a hard time seeing how to approach this. Given $\phi : \Bbb R^x \rightarrow \Bbb R^x$ is an automorphism of $\Bbb R^x$ (the multiplicative group of nonzero real numbers), and $P$ = {$x \in ...
0
votes
1answer
32 views

About the special linear group $SL(n,\,\mathbb{Z})$

I found the following relation: $$SL_n(\mathbb{Z})=\langle e+e_{12},\, e_{12}+\dots+e_{n-1, n}+(-1)^{n-1}e_{n1}\rangle$$ where $e_{ij}$ is the matrix with its $(i,\, j)$th entry $1$, and all other ...
0
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2answers
15 views

Question about the centralizer and conjugacy classes

Let $G$ be a finite $p$-group and $H$ a non trivial normal subgroup of $G$. I want to prove that $H\cap Z(G)\neq 1$. I define a relation in $H$ by $x\sim y$ if and only if there exists $g\in G$ such ...
0
votes
1answer
17 views

injectivity. identity map of a ring to its tensor product

Let $B$ be an $A$-algebra, $f: B \rightarrow B \otimes_{A} B$ is defined by $f(b)=b\otimes 1$. Is $f$ injective? I know the definition of tensor product and started from representing as $(b,1)=\sum ...
0
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2answers
588 views

Proving the Heisenberg Group is a group

I have to prove that the Heisenberg Group, \begin{pmatrix} 1&a&b \\ 0&1&c\\ 0&0&1 \end{pmatrix} where $a,b,c\in\mathbb{R}$ is a group. I am proving a group ...
0
votes
1answer
34 views

Finding the irreducible polynomial of $\zeta_6, \zeta_8, \zeta_9$ over the field $\mathbb{Q}(\zeta_3).$

How can I get started on finding the irreducible polynomial of $\zeta_6, \zeta_8, \zeta_9$ over the field over $\mathbb{Q}(\zeta_3)?$ Should I construct field extensions and then use the degrees of ...
3
votes
3answers
47 views

Integral Domains and Unique Factorisation Domains

I'm learning about Rings, commutative rings, IDs, UFDs, etc with each being a subset of the predecessor, and I'm now trying to find an ID that is not a UFD I understand $\mathbb Z[\sqrt{-5}]$ is an ...
2
votes
2answers
50 views

Ideal in $\mathbb Z[x]$ which is not two-generated

I can prove that the ideal $(4, 2x, x^2)$ in $\mathbb Z[x]$ is not principal. But I failed to prove that this cannot be generated by two elements. It's really difficult for me. Would you give me a ...
0
votes
0answers
51 views

When this operation is associative?

I am looking for all possible positive values of $\alpha$ such that the binary operation (on natural numbers) defined by $$m\circ n = mn + \lfloor\alpha n\rfloor\lfloor\alpha m\rfloor$$ is ...
-1
votes
0answers
27 views

the definition of Semigroups as S[x] [on hold]

how is the definition of S[x]? In rings it is said to be polynomial but in semigroup what is like that?
2
votes
1answer
46 views

Subring of a commutative Noetherian ring

We know that it's possible subring of the commutative Noetherian ring become not Noetherian (for example: Subring of a Noetherian ring need not be Noetherian?). But if $S$ be a subring of ...
0
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2answers
34 views

Show the two fundamental groups representation of the Klein bottle are the same

Show the two different representation of the fundamental groups of the Klein bottle $G=<a,b:aba^{-1} b>$ and $H=<c,d:c^2 d^{-2}>$. As we calculate in different and gett one result as ...
0
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1answer
34 views

$R/Ra$ is an injective module over itself

Let $R$ be a PID, $a\in R$ be a nonzero nonunit in $R$. Prove that $R/Ra$ is an injective module over itself. If $R$ is a PID, every $R$- divisible module is injective, but the question concerns ...
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0answers
22 views

Properties of divisible modules

I've got two questions concerning divisible modules that are defined as follows: Let $M$ be a $R$-module. $M$ is called divisible iff for all $r\neq 0$ the map $\phi_r :M \to M $ $m\mapsto rm$ is ...
1
vote
1answer
19 views

Multiple Group Representations using Cayley's Thm

I know that an abstract group can be made isomorphic to a subgroup of a symmetric group, by using a Cayley table for that abstract group. However, what is a technique for getting another permutation ...
2
votes
2answers
34 views

Nilpotent elements in the quotient ring of a polynomial ring

If $F$ is a field and $p(x) \in F[x]$, prove that the ring $R=F[x]/(p(x))$ has no nonzero nilpotent elements iff $p(x)$ is not divisible by the square of any polynomial. (==>) $R$ has no ...
0
votes
2answers
43 views

Why $K = (X_1, X_2, …)$, the ideal generated by $X_1, X_2, …$ not finitely generated as a R-module?

Let $R = \mathbb{Z}[X_1, X_2, \dots]$ be the ring of polynomials in countably many variables over $\mathbb Z$. Why $K = (X_1, X_2, ...)$, the ideal generated by $X_1, X_2, ...$ is not finitely ...
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votes
0answers
29 views

problem about p-cycles [on hold]

Let a be a cycle in $S_n$ so that $a\neq(1)$ And $a^p=(1)$ with $n/2<p\leq n$ and $p$ being prime. Prove that $a$ is a p-cycle.
0
votes
2answers
42 views

is 0 in the following Ideal?

Given $R=\mathbb R[x]$ and $I=(2x^3-3x^2+2x-3)+(2x^2-x-3)$ Is an Ideal of R? I don't understand what the quantity I is... Am I supposed to sum them together giving $2x^3-x^2+x-6$ Now here's the ...
9
votes
1answer
164 views

Why are “algebras” called algebras?

There's a mathematical object called an "algebra" (e.g. an algebra over a ring), but why does this particular object have such an "important" name (which makes it sound like the most important concept ...
3
votes
2answers
49 views

Galois group of $x^4-2$

I am trying to explicitly compute the Galois group of $x^4-2$ over $\mathbb{Q}$. I found that the resolvent polynomial is reducible and the order of the Galois group is $8$ using the splitting field ...
0
votes
1answer
19 views

are calculations in ideals associative?

Say if we have $I+a+bX+cX^2+I+I$, can we rearrange the order to how we like? Because you can always imagine $0+I$ when the ideals are written consecutively.
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0answers
28 views

Field extension in rational functions [duplicate]

I'm facing the following problem: Let $ F $ be a field, and let $ F(x) $ denote all rational functions over $ F $ (functions of form $\frac{P(x)}{Q(x)}$, where $ P,Q$ are polynomials over $ F $). ...
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1answer
36 views

About ring $( \frac{F[t]}{t^2})$

I was trying to study some properties of the group $GL_2 ( \frac{F[t]}{t^2})$ where $F$ is a field with $p$ elements. $p$ is a prime. I found that $ \frac{F[t]}{t^2}$ is a discrete Valuation ring. ...
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2answers
20 views

How to get an equivalent permutation [closed]

What is the other form of the permutation $\sigma=(12)(345)$? how do I solve this kind of problem?
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1answer
29 views

Primary Decomposition Theorem applied to projections in $\mathbb{R}^2$

We have recently learned the Primary Decomposition Theorem in my Algebra course. I came up with what I think is an instance of this theorem but I haven't convinced myself. Supposed we have the map ...
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1answer
40 views

show that $g(x)= x^n f(\frac 1x) \in \Bbb F[x]. $ where $\Bbb F $ is a field

Let $\Bbb F $ be a field and $f(x)=\sum_0^n a_i x^i \in \Bbb F[x]$. Show that $g(x)= x^n f(\frac 1x) \in \Bbb F[x]$ Show that if $r \neq 0$ is a root of $f(x)$ then $r^{-1}$ is a root of $g(x)$ Find ...
0
votes
1answer
40 views

I have to show $p=p(x-\lambda)$ if and only if they have the same zeros in $F$

Suppose $F$ is a field, $|F|\geq n \geq 2$. Given $\lambda \in F$ I know $p,p(x-\lambda)\in F[x]$ are irreducible monic polynomials. I have to show $p=p(x-\lambda)$ if and only if they have the same ...
0
votes
1answer
25 views

Generators of a subgroup of the modular group

I am looking to show the following: If we consider the group of 2 by 2 matrices generated by $S$=$\begin{bmatrix}1 & 0\\2 &1\end{bmatrix}$ and $T$=$\begin{bmatrix}1 & 2\\0 ...
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2answers
27 views

I have to show $p(x-\lambda)$ is an irreducible monic poynomial.

Suppose that $\mathbb{F}$ is a field , $|\mathbb{F}|\geq n \geq 2$. I know $p\in \mathbb{F}[x]$ is an irreducible polynomial and let $\lambda \in \mathbb{F}$ , I have to show $p(x-\lambda)$ is an ...
0
votes
2answers
30 views

Stabilizers of permutations

The problem is I need to list all elements in the stabilizer of the permutation $\sigma=(123)$ in $S_5$. I know that the stabilizer of a permutation $\sigma$ is the subgroup $$Stab_{{S_n}}(\sigma) = ...
2
votes
1answer
42 views

If $N_1, N_2, \dots, N_n$ are distinct unique Sylow subgroups, how do we know $N_i \cap (N_1N_2 \cdots N_{i-1}N_{i+1} \cdots N_n)=\{1\}$?

If $N_1, N_2, \dots, N_n$ are distinct unique Sylow subgroups, how do we know $$N_i \cap (N_1N_2 \cdots N_{i-1}N_{i+1} \cdots N_n)=\{1\}$$
3
votes
1answer
15 views

unknown permutation [duplicate]

Can someone help on this. I'm stuck on this part. I am trying to find a permutation $\sigma$ such that $$\sigma(1,2)(3,4)\sigma^{-1} = (5,6)(3,1)$$. By a particular theorem, I know I can have this one ...
1
vote
1answer
49 views

If $\left<3\right> \cong \mathbb Z_2$ and $\left<2\right> \cong \mathbb Z_3$, why isn't $\left<3\right>\left<2\right> \cong \mathbb Z_2\mathbb Z_3$

$\left|\mathbb Z_{6}\right|=6=2 \cdot 3$. Since $\mathbb Z_{6}$ is abelian, all subgroups are normal and thus its Sylow subgroups are unique. $\left<3\right>$ is the $2$-Sylow, and ...