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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Exercise 2.26 Atiyah-Macdonald, flatness

I'm stuck on this exercise. $A$ is a commutative ring with unit. $N$ is an $A$-module. Then $N$ is flat $\Longleftrightarrow $ $\text{Tor}_{1}(A/a, N ) = 0 $ for every finitely generated ideal $a$ of ...
2
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1answer
135 views

Endomorphism Ring Isomorphism

Suppose that $D$ is a division ring and let $M_n(D)$ be the $n\times n$ matrix ring with entries from $D$. $D^n$ is a left module over $M_n(D)$. I want to show that $D$ and End$_{M_n(D)}(D)$ are ...
3
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4answers
224 views

General approach for finding how many group homomorphisms are there

So I've asked this type of questions for more than once, and still I don't get the method(s) I've been presented with. What's the general recommended method for finding how many homomorphisms are ...
0
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1answer
47 views

Show that two field extensions are the same

Can you help me with showing that these two field extensions are the same: $\mathbb{Q}(\sqrt{3}, \sqrt[3]{5})$ $\mathbb{Q}(\sqrt{3} + b\sqrt[3]{5})$, where $b\neq0$ is any rational number. Thanks ...
4
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1answer
200 views

Exercise 2.11 Atiyah-Macdonald

This is part of the exercise, I'm stuck with it. $A$ is a commutative ring with unit. 1) Suppose we have an homomorphism $\phi : A^{m} \to A^{n}$ surjective. Is true that $m \geq n $ ? 2) Suppose ...
6
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2answers
207 views

A non-abelian group such that $G/z(G)$ is abelian.

I'm looking for an example of a non-abelian group $G$ such that $G/z(G)$ is abelian, where $z(G)$ is the center of the $G$. In other words, I'm looking for a non-abelian group where $z(G)$ contains ...
2
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1answer
362 views

Free group of finite rank: subgroup of finite index

This is a well-known result, but I can't find a proof of it, without using topology. Let $m\geq2$ be an integer. Then the free group of rank $2$ contains the free group of rank $m$ as a finite-index ...
3
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3answers
145 views

Check that $\sqrt{7}(\sqrt[3]{5} - \sqrt[5]{3})$ is not rational

How to prove that $\sqrt{7}(\sqrt[3]{5} - \sqrt[5]{3})$ is not rational. I will appreciate any proof, but I had such exercise during lecture in field theory. Thanks.
1
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1answer
91 views

Find a subfield of $\mathbb{C}$ isomorphic to other field

Do you know, how I can find a subfield of $\mathbb{C}$ isomorphic to $F = \mathbb{Q}(\sqrt[3]{7})$ such that $F \nsubseteq \mathbb{R}$? I don't even have clue, how I should start. Thanks
0
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1answer
71 views

Equality with powers of an ideal

Let $A$ be an arbitrary (commutative with an identity) ring. Suppose $\alpha$ is an ideal. Is it true that $$\alpha(\alpha\cap\alpha^2\cap\alpha^3\cap…)=\alpha\cap\alpha^2\cap\alpha^3\cap…?$$ ...
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3answers
83 views

Let G be a finite group that has elements of every orders from 1 - 12. What is the smallest possible value of |G|

Let G be a finite group that has elements of every orders from 1 through 12. What is the smallest possible value of |G| I need your help
0
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2answers
87 views

What is the cyclic subgroup generated by $9$ in $U(28)$

$U(28)=\{1,3,5,9,11,13,15,17,19,23,25,27\}$ So $<9>=\{1,9,25\}$, correct? Why is this the case? How is it generating $25$? Wouldn't $27$ be the last element?
2
votes
1answer
205 views

Endomorphisms of direct sum and division ring

How to prove that $$\operatorname{End}_R(V^{ \oplus n }) \cong M_n(D),$$ where $V$ is a simple left $R$-module and $D=\operatorname{End}_R(V)$. This is part of the proof finally leads to prove ...
0
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1answer
30 views

Understanding units in $Z_{12}$

In $Z_{12}$ , the elements $1$, $5$, $7$, and $11$ are units. But how and why? I can't see the connection between $1$, $5$, $7$, and $11$ and how they have multiplicative inverses in $Z_{12}$ The ...
4
votes
2answers
135 views

Why is $Z_2 \times Z_4$ not isomorphic to $Z_8$?

$Z_2 \times Z_4$ has 8 elements and obviously so does $Z_8$. How do you conclude they are not isomorphic then? They essentially have a $1-1$ relationship
2
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1answer
24 views

What does the statement “The cyclic subgroup of $Z_{24}$ generated by $18$ has order $4$” mean?

The group $Z_{24}$ has 24 elements. But $18$ can't generate all those elements. The best it can do is $18+18=12$ $18+18+18=6$ $18+18+18+18=0$ $18+18+18+18+18=18$ $18+18+18+18+18+18=12$ What am I ...
0
votes
1answer
79 views

Short Exact Sequences

Let $M \ge N \ge P$ be R-modules. Prove that there exist natural (not depending on choices) R-homomorphisms $N/P \to M/P$ and $M/P \to M/N$ for which the sequence $0 \to N/P \to M/P \to M/N \to 0$ is ...
1
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1answer
62 views

Exercise from Atiyah about flatness

This is an exercise from Atiyah. Let $N$ be a flat $B$-module, and $B$ a flat $A$-algebra where $A$ is a commutative ring with unit. Then $N$ is flat as $A$-module Any hint ?
2
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2answers
49 views

How can I show that $G/[A,B]$ is an abelian group?

$G$ is a group such that $G=AB$, where both $A$ and $B$ are abelian. We denote $G':=\langle[g,h]\;:\;g,h\in G\rangle=[G,G]$ and similarly $[K,H]:=\langle[k,h]\;:\;k\in K,\;h\in H\rangle$ for $H, K$ ...
2
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0answers
74 views

Describe up to isomorphism the semidirect product

I have the following problem: I have to describe up to isomorphism the semidirect product $C_6 \rtimes C_2$, where $C_6$ denotes cyclic group of order six. I think I have to use external semidirect ...
7
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1answer
83 views

$G\times G\cong H\times H\Longrightarrow G\cong H$ for $G$ ACC and DCC

I want to prove that, if $G$ satisfies ACC and DCC on normal subgroups, then $G\times G\cong H\times H$ implies $G\cong H$. I observed that if we can prove that $G\times G$ satisfies ACC and DCC the ...
0
votes
1answer
53 views

kernel of k-homomorphism

Let's $f_1,\ldots,f_m \in k[x_1,\dots,x_n]$ and the $k$-algebra homomorphism: $$g:k[x_1,\ldots,x_n,y_1,\ldots,y_m] \rightarrow k[x_1,\ldots,x_n]$$ that sends $y_i \mapsto f_i$ and $x_j \mapsto x_j$ ...
1
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0answers
123 views

Any non-trivial finitely-generated group admits maximal subgroups

I want to solve the following problem from Dummit & Foote's Abstract Algebra: This is exercise involving Zorn's Lemma (see Appendix I) to prove that every nontrivial finitely generated group ...
0
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1answer
141 views

Maximal ideal generated by irreducible element

Let $R$ be an integral domain and let $(c)$ be a non-zero maximal ideal in $R$. Prove that $c$ is an irreducible element.
0
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1answer
44 views

nontrivial $K$-automorphism of $K(x)$

How can I find $K$-automorphism $\sigma \in Aut(K(x))$ different from identity such that $\sigma (x(x+1))=x(x+1)$?
32
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6answers
2k views

Can you give me some concrete examples of magmas?

I've seen the following: I've learned a bit about groups and I could give examples of groups, but when reading the given table, I couldn't imagine of what a magma would be. It has no ...
2
votes
2answers
154 views

Why define $\gcd(r,s)$ as a positive generator $d$ of the cyclic group $H=\{nr+ms|n,m\in\mathbb{Z}\}$?

This is in regards to definition 6.8, p. 62 from Fraleigh's "A first course in abstract algebra". 6.8 Definition Let $r$ and $s$ be two positive integers. The positive generator $d$ of the ...
2
votes
2answers
34 views

Let H be a subgroup of $G$, then the center of $H$ is a subgroup of $Z(G)$.

It is a true or false problem and answer is said to be False but I'm trying to figure out how it is so. All I know is that, by definition the center of a group $G$ is: $Z(G)$={$g\in G|gx=xg$ , for ...
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2answers
81 views

Prove that the number of subgroups in $D_n = \tau (n) + \sigma (n)$

Prove that the number of subgroups in $D_n = \tau (n) + \sigma (n)$ where $\tau (n)$ represents number of divisors of $n$ and $\sigma (n)$ represnts the sum of divisors of $n$. Attempt: $D_n = ...
3
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1answer
45 views

Conjugate subgroups are normal?

Suppose $A \subset C$ and $B \subset C$. Assume $A$ and $B$ are conjugate subgroups, that is $cAc^{-1}=B$ for some $c \in C$. Is the following statement true? $A=B$ if and only if $A$ and $B$ are ...
2
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1answer
38 views

Factor groups and Burnside's lemma

I'm supposed to find the number of orbits in $\{1,2,3,4,5,6,7,8\}$ under the cyclic subgroup generated by $(1,3,5,6)$ of $S_8$. I would have very much appreciated an explanation to this exercise since ...
0
votes
2answers
128 views

Finding the 8 Automorphisms of $\mathbb{Q}[\sqrt[4]{2}, i]$

I think I will have i with root that identity ,then -the root,-i with root ,the root alone I cannot understand finding 8 ??
4
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2answers
71 views

$G$ an abelian group, $n>1$ a fixed integer, and $\phi :G\to G$ defined by $\phi(a)=a^n$ for $a\in G$. Determine wheter $\phi$ is onto.

$G$ an abelian group, $n>1$ a fixed integer, and $\phi :G\to G$ defined by $\phi(a)=a^n$ for $a\in G$. Determine wheter $\phi$ is onto. I think it totally depends on different situations. ...
2
votes
1answer
245 views

Sum and product of two transcendental numbers can't be both algebraic

Suppose $a$ and $b$ are complex numbers and both transcendental over $\mathbb Q$. I am wondering why $ab$ and $a+b$ can not both be algebraic. Thanks for any help.
3
votes
1answer
142 views

All finite abelian groups of order 1024

List all finite abelian groups of order 1024. Attempt: The prime decomposition of 1024 is 1024 = 2^10. So Z_1024 = Z_2 x Z_512 ...
3
votes
1answer
131 views

In a Morita context ($A, B, P, Q, f, g$), why are $P$ and $Q$ projective if $f$ is surjective?

The title probably says it all :). This is probably a very very simple question, please bear with me. Let $(A, B, P, Q, f, g)$ be a Morita context (or pre-equivalence data) as defined by Hyman Bass in ...
1
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1answer
65 views

Rank of finitely generated abelian groups

Is it true that for every f.g. abelian group $A$ we have $\mathrm{rank} \ A=\dim_{\mathbf{R}}\mathrm{Hom}_{\mathbf{Z}}(A,\mathbf{R})$? (Here the vector space structure is defined in a natural way. I ...
12
votes
2answers
536 views

Is number rational?

How can we check if number $a=\frac{ \sqrt[4]{2}+\sqrt[3]{3}}{\sqrt[4]{2}+\sqrt[3]{3} +1}$ is rational? Is there any smart solution? Another assignment is to find $\left( ...
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1answer
31 views

How to deal with equal fractions in polynomial quotient ring Q[x]

I have an equation $\frac{p}{q}=\frac{r}{s}$ in Q[x] with the known property that the gcd of p and q is 1. I would like to infer $r=p$ and $s=q$ but doubt that this is correct. ( I already know from ...
3
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1answer
65 views

Classify abelian groups $A$ which are irreducible $End(A)$-modules

Classify abelian groups $A$ which are irreducible $End(A)$-modules. I think i did it for finite abelian group $A$ . A finite abelian group $A$ is irreducible iff order of $A$ a is power of prime. ...
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0answers
28 views

Descending chain condition on a finite dimension algebra

In a proof I'm reading, the author says "As $A$ is finite dimensional, a descending chain of left ideals must stabilize." The context is that $A$ is a finite dimensional simple $k$-algebra i.e. it ...
3
votes
2answers
97 views

Let $H,K \trianglelefteq G$. Show that if $H$ and $K$ are solvable then the subgroup $HK$ is solvable.

Let $H,K \trianglelefteq G$. Show that if $H$ and $K$ are solvable then the subgroup $HK$ is solvable. I did the following: Since H is solvable, there is a subnormal series $\{e\} = H_0 ...
0
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2answers
160 views

Spectrum of a product of rings isomorphic to the product of the spectra

I've found in an exercise this statement: If $A$ is a commutative ring with unit and $A = A_{1} \times \dots \times A_{n}$ then $$\def\Spec{\operatorname{Spec}} \Spec(A) \cong \Spec(A_{1})\times ...
0
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1answer
61 views

$\gcd (f,g)$ is a non-trivial divisor of $f$

Suppose that $\pi$ is an irreducibel polynomial and $\pi$ divides $f$ and $g$ where $0<\deg(g)<\deg(f)$. Why is $\gcd(f,g)$ then a non-trivial divisor in $f$ ? Maybe I should mention that ...
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1answer
89 views

Fraction field of $R/P$

It may be a simple question seeming too easy, but I seek a help: If $P$ is a prime ideal of a commutative ring $R$, could one say that $R_P/PR_P$ is the field of fractions of $R/P$? Thanks a ...
5
votes
1answer
122 views

Diagonal morphism

Hello I'm starting to study algebraic geometry on my own with a book and I've been thinking on this problem a few days. I'd appreciate if someone could help me. Let $V$ be an affine variety. We ...
2
votes
2answers
38 views

Show that if $p$ is a prime of the form $p=4n+1$, then we can solve $x^2\equiv -1\mod p$(with $x$ an integer).

Show that if $p$ is a prime of the form $p=4n+1$, then we can solve $x^2\equiv -1\mod p$(with $x$ an integer). My attempt:If $p$ is a prime, then $U_p=${$[x]|1\leq x<p$} is cyclic.
2
votes
1answer
77 views

Image and Kernel of different linear maps and their dimension [closed]

I'm trying to determine the image and the kernel of different linear maps. I understood well the theory but I can not transfer the knowledge of the books I have read to specific linear maps. 1) ...
0
votes
1answer
63 views

Adjoining elements to a ring.

I have a question in Artin's Algebra, about adjoining elements on page 339. Proposition 11.5.5 says: Let $R$ be a ring, and let $f(x)$ be a monic polynomial of positive degree $n$ with coefficients ...
1
vote
1answer
75 views

Prove that the center of group G is a subgroup of G. [duplicate]

By definition, the center of G is the set: $Z(G)$ = {$g\in G|g^{-1}xg=x$ for all $x\in G$} We need to show that: The identity element exists It is closed under the operation For every element $g$, ...