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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.

2
votes
1answer
107 views

Isomorphy of quotient modules implies isomorphy of submodules

Let A be some commutative ring, $M$ an $A$-module and $N_1, N_2$ two submodules of $M$. If we have $M/N_1 \cong M/N_2$, does this imply $N_1 \cong N_2$? This seems so trivial, but I just don't see a ...
30
votes
8answers
13k views

Why $\gcd(qb+r,b)=\gcd(b,r)$?

Given: $a = qb + r$. Then it holds that $\gcd(a,b)=\gcd(b,r)$. That doesn't sound logical to me. Why is this so? Addendum by LePressentiment on 11/29/2013: (in the interest of http://meta.math....
9
votes
3answers
4k views

Dedekind domain with a finite number of prime ideals is principal

I am reading a proof of this result that uses the Chinese Remainder Theorem on (the finite number of) prime ideals $P_i$. In order to apply CRT we should assume that the prime ideals are coprime, i.e. ...
4
votes
1answer
2k views

$\mathbb{Q}$ is an injective $\mathbb{Z}$-module

I just learned what an injective module is and I want to consider some basic examples. Apparently, $\mathbb{Q}$ is an injective module over $\mathbb{Z}$, but I can't find an elementary proof of this ...
2
votes
0answers
136 views

First Isomorphism Theorem in Generality [duplicate]

Possible Duplicate: Quotient objects, their universal property and the isomorphism theorems In how general of a setting is the first isomorphism theorem true? Wikipedia states it in terms of ...
2
votes
1answer
534 views

concrete isomorphisms of polynomial rings

Question 1: In A Singular Introduction to Commutative Algebra, page 222, there is written: How can I check that this isomorphism actually holds? I would really prefer a computational proof (using a ...
15
votes
4answers
1k views

Are free products of finite groups virtually free?

Is the free product $A*B$ of two nontrivial finite groups always virtually free? If yes, is it easy to show?
9
votes
5answers
2k views

Commutative rings without assuming identity

I was going through Exercises in Dummit&Foote, which does not assume identity in the definition of a ring, and reached the following exercise: Prove that in a Boolean ring ($a^2 = a$ for all $a$...
3
votes
2answers
6k views

Necessary and Sufficient Condition for a sub-field

Is there any necessary and sufficient condition to determine whether a subset $H$ of a given field $K$ is a subfield? In some paper I have found something like that: $H$ is a field if for all $a, b\...
1
vote
1answer
270 views

How to show some systems of equations do not have a closed form solution?

How to show some systems of equations do not have a closed form solution? for example I was once given something similar to ( this might not be the exact problem but i am just using it as an example ...
6
votes
1answer
3k views

Splitting field implies Galois extension

I hope this isn't too elementary of a question, but I'm not sure I understand Artin's proof that if $K/F$ is a finite extension, then $K/F$ Galois is equivalent to $K$ being a splitting field over $F$ ...
7
votes
2answers
557 views

Every ideal of $K[x_1,\ldots,x_n]$ has $\leq n$ generators?

Is this true: Every ideal of $K[x_1,\ldots,x_n]$ is generated by some subset with $\leq n$ elements? It is true when $n=1$, since $K[x]$ is a PID. I'm trying to prove it is not true for $n\geq2$, ...
9
votes
2answers
6k views

Irreducible in $\mathbb{Z}[\sqrt{-5}]$

How can I prove that $2+\sqrt{-5}$ is irreducible in $\mathbb{Z}[\sqrt{-5}]$? I tried to show by $2+\sqrt{-5}=(a+b\sqrt{-5})(c+d\sqrt{-5})$ but I could not get a contradiction.
12
votes
3answers
4k views

If a ring is Noetherian, then every subring is finitely generated?

Let $R$ be a commutative ring with $1$, and let $K$ be a field. We know that $R$ is Noetherian iff every ideal of $R$ is finitely generated as an ideal. Question 1: If $R$ is Noetherian, is every ...
2
votes
1answer
268 views

Dedekind domains

Let $A$ and $B$ be ideals. I want to show that there exists an element $c \in K$ (where $K$ is the quotient field of a Dedekind domain $O$) such that $cA$ is an ideal relatively prime to $B$. As ...
2
votes
3answers
154 views

Question about Notation

What does $\mathbb{R}∗\mathbb{R}$ mean? I'm sure this has been asked before, but I do not know how to search for notations in past questions.
0
votes
3answers
5k views

What do these terms mean: commutative, associative, distributive

I am reading a book, and I am trying to understand what the writer really mean by the following terms. I would like to understand what these words mean in relation to the examples. In regular algebra,...
3
votes
5answers
2k views

How to show that $p(x)$ divided by $x-c$ has remainder $p(c)$?

This is from Pinter, A Book of Abstract Algebra, p.265. Given $p(x) \in F[x]$ where $F$ is a field, I would like to show that $p(x)$ divided by $x-c$ has remainder $p(c)$. This is easy if $c$ is a ...
13
votes
1answer
406 views

Working with Morphisms in Local Coordinates

In light of the holiday, I would like to air a grievance. I have no good way to recoordinatize a morphism of varieties as I move between coordinate neighborhoods. Let me explain what I mean with ...
1
vote
2answers
602 views

How to find the number of homomorphisms of $\mathbb{Z}$ into $\mathbb{Z}$?

Consider the maps $\mu:\mathbb{Z}→\mathbb{Z}$ and $\mu:\mathbb{Z}→\mathbb{Z}_2$. For example if I am asked to find the number of homomorphisms of $\mathbb{Z}$ into $\mathbb{Z}$, and of $\mathbb{Z}$ ...
3
votes
1answer
358 views

Computing with ideals: over $K$ or over $\mathbb{Q}\subseteq K$? does it matter? pt.2

Definitions: Let $R$ be any commutative ring with $1$, and let $S\subseteq R$ and $I\unlhd R$. The ideal generated by $S$ is $\langle S\rangle_R:=\bigcap\{J; S\subseteq J\unlhd R\}$, i.e. the ...
2
votes
1answer
3k views

Why is the determinant a homomorphism? [duplicate]

Possible Duplicate: How to show $\det(AB) =\det(A)\det(B)$ Consider the map $q:\operatorname{GL}(n,R)→R^*$ given by $q(A)=\det(A)$. I know that $$q(AB)=q(A)q(B)=\det(A)\det(B)$$ does not hold if ...
21
votes
1answer
2k views

Is anybody researching “ternary” groups?

As someone who has an undergraduate education in mathematics, but didn't take it any further, I have often wondered something. Of course mathematicians like to generalize ideas. i.e. it is often ...
2
votes
2answers
1k views

Group of order $105$ has a subgroup of order $21$

I am trying to prove that a group of order $105$ has a subgroup of order $21$. I know it can be done using Sylow theorems, I was just wondering if the proof below could be another way of doing that. $...
0
votes
3answers
651 views

Determine a generator $g$ of the group $\mathbb{Z}_p^\times$ given some elements and their inverses

Is there a way of finding a generator of a multiplicative group $G = \mathbb{Z}_{41723027}^\times$ given some elements of the group : S = $\{ 4, y=1063, 1064, y^{-1}=12049830, 41723026 \}$ In the ...
6
votes
2answers
221 views

Why do we assume $a+b=b+a$ in a Ring with 1. Also, is it true with Rings without 1?

I was wondering why do some people use redundant axioms in definitions? If you just expand $(a+1)(b+1)=(a+1)b+a+1=ab+b+a+1$ $(a+1)(b+1)=a(b+1)+b+1=ab+a+b+1$. Hence, $ab+a+b+1=ab+b+a+1$, then cancel ...
17
votes
3answers
2k views

Proof that no permutation can be expressed both as the product of an even number of transpositions and as a product of an odd number of transpositions

I am aware that there are a couple of well-known proofs of this theorem, but I'm specifically grappling with the proof given in Fraleigh's A First Course in Abstract Algebra (Theorem 9.15 in the ...
4
votes
3answers
412 views

Symmetric group [duplicate]

Possible Duplicate: What kind of “symmetry” is the symmetric group about? Could you tell me please, why Symmetric group is called "symmetric"? I found an example with quadrate, where ...
9
votes
5answers
4k views

Proving that $G/N$ is an abelian group

Let $G$ be the group of all $2 \times 2$ matrices of the form $\begin{pmatrix} a & b \\ 0 & d\end{pmatrix}$ where $ad \neq 0$ under matrix multiplication. Let $N=\left\{A \in G \; \colon \; ...
0
votes
1answer
102 views

To show $\mathcal{L}(N)$ is a normal subgroup of $G'$

Let $G$ be group, $N$ is normal in $G$ and $\mathcal{L}:G \to G^\prime$ be a surjective group homomorphism, then prove that the image $\mathcal{L}(N)$ of $N$ is a normal subgroup of $G’$. I let $g_{...
3
votes
3answers
461 views

Existence of subgroup of order six in $A_4$

Show that the alternating group $A_4$ of all even permutations of $S_4$ does not contain a subgroup of order $6$. For me am thinking to write all elements of $A_4$ and trying to find every ...
1
vote
1answer
805 views

How to find kernel of this map: is my attempt correct?

Find $\ker(\varphi)$ and $\varphi(3,10)$ for $\varphi \colon \mathbb Z\times\mathbb Z\to S_{10}$ such that $\varphi(1,0)=(3,5)(2,4)$ and $\varphi(0,1)=(1,7)(6,10,8,9)$. For example if I get order ...
0
votes
2answers
177 views

Somebody help me how to find $\ker(Q)$ and $Q(21)$

In my study of group homomorphism I find a question like this. Let $Q\colon \mathbb Z\to S_8$ be group homomorphism such that $Q(1)=(1,4,2,6)(2,5,7)$. Then find $\ker(Q)$ and $Q(21)$. In my mind I ...
6
votes
1answer
1k views

Proof of Yoneda Lemma [closed]

Can anyone explain to me Yoneda Lemma proof in great details? i.e. they usually say " ... it is easy to see that these morphisms are inverse to each other.." without explanation.
0
votes
2answers
112 views

How to find the elements of $S_n$?

Consider $S_n$, I do not understand how to get the elements of $S_n$ especially for $n \geq 4$ Any one to show me how simply I can find them. Thanks a lot
9
votes
2answers
507 views

Which groups have precisely two automorphisms

Which groups $G$ have precisely two automorphisms, i.e., precisely one non-trivial automorphism? Examples: $G= C_3, \mathbf{Z},\ldots$. I think $G$ has to be abelian. In fact, we have $ \vert G\vert ...
1
vote
1answer
200 views

Does there exist a group without automorphisms such that…

Does there exist a non-trivial group $G$ without automorphisms, a homomorphism $f:H\to G$ and a homomorphism $g:G\to H$ such that $g\circ f = \mathrm{id}_H$ for some group $H$ with non-trivial ...
9
votes
4answers
790 views

Relationship between algebraically closed fields and complete metric spaces?

I've been reading recently about algebraically closed fields and complete metric spaces, and it seems to me that they are very similar ideas. Is there some more general mathematical concept which ...
16
votes
2answers
423 views

Non-algebraically closed field in which every polynomial of degree $<n$ has a root

My problem is to build, for every prime $p$, a field of characteristic $p$ in which every polynomial of degree $\leq n$ ($n$ a fixed natural number) has a root, but such that the field is not ...
11
votes
2answers
859 views

Elementary proof of $\mathbb{Q}(\zeta_n)\cap \mathbb{Q}(\zeta_m)=\mathbb{Q}$ when $\gcd(n,m)=1$.

In an answer to another question I used the fact that $\mathbb{Q}(\zeta_m)\subseteq \mathbb{Q}(\zeta_n)$ if and only if $m$ divides $n$ (here $\zeta_n$ stands for a primitive $n$th root of unity, Edit:...
14
votes
3answers
3k views

Software for Galois Theory

Background: While studying Group Theory ( Open University M208 ) I had a lot of benefit from the Mathematica Add-on package AbstractAlgebra and later from the GAP software. I am currently self-...
3
votes
2answers
1k views

Dihedral group generators.

For convenience, consider the symmetries of a square. When finding/specifying the reflection generator of a dihedral group does it matter which relfection I choose? Dummit/Foote say to choose the ...
3
votes
1answer
184 views

Counting endomorphisms of $\mathbf Q(\zeta _{n})$

If $\zeta= \zeta_{n}$, how does one count the homomorphisms $f:\mathbf{Q}(\zeta)\rightarrow \mathbf{Q}(\zeta)$?
2
votes
1answer
94 views

For $n\ge 3, x_{1},…,x_{n} \in \mathbf{Q}^{\ast}$, $[\mathbf{Q}(\sqrt{x_{1}},…\sqrt{x_{n}}) : \mathbf{Q}] < 2^{n}$

For $n\ge 3, x_{1},...,x_{n} \in \mathbf{Q}^{\ast}$ and $[\mathbf{Q}(\sqrt{x_{1}},...\sqrt{x_{n}}) : \mathbf{Q}] < 2^{n}$ how can we conclude that there are non empty $I \subset \{1,...,n\}$ with $...
4
votes
1answer
810 views

If $R$ is an integral domain, and every $R$-module is projective, must $R$ be a field?

Let $R$ be an integral domain with the property that all modules over $R$ are projective. Does it follow that $R$ is a field? Obviously the converse is true.
1
vote
3answers
2k views

Intuitive understanding regarding the number of roots of a polynomial over a field and invertibility

In considering: A polynomial over a field of degree n has at most n roots. -- How does this make use of the stipulation "over a field" - especially with an eye toward inveritbility ? Is it to insure ...
7
votes
5answers
3k views

If an Abelian group $G$ has order $n$ and at most one subgroup of order $d$ for all $d$ dividing $n$ then $G$ is cyclic

If an Abelian group $G$ has order $n$ and at most one subgroup of order $d$ for all $d$ dividing $n$ then $G$ is cyclic. I am trying to use the structure theorem for finitely generated abelian groups....
5
votes
2answers
2k views

Normal sylow subgroup of a finite group is characteristic

Let $P$ be a normal sylow $p$-subgroup of a finite group $G$. Since $P$ is normal it is the unique sylow $p$-subgroup. I would like to say if $\phi$ is an automorphism then $\phi(P)$ is also a sylow ...
5
votes
1answer
242 views

For which $n\in\mathbf{N}$ do we have $\mathbf{Q}(z_{5},z_{7}) = \mathbf{Q}(z_{n})$?

Put $z_{n} = e^{2\pi i /n}$. I am searching for $n \in \mathbf{N}$ so that $\mathbf{Q}(z_{5},z_{7}) = \mathbf{Q}(z_{n})$. I know that : $z_{5} = \cos(\frac{2\pi}{5})+i\sin(\frac{2\pi}{5}) $ and $z_{7}...
62
votes
5answers
30k views

Is $\mathbf{Q}(\sqrt{2}, \sqrt{3}) = \mathbf{Q}(\sqrt{2}+\sqrt{3})$? [duplicate]

Is $\mathbf{Q}(\sqrt{2}, \sqrt{3}) = \mathbf{Q}(\sqrt{2}+\sqrt{3})$ ? $$\mathbf{Q}(\sqrt{2},\sqrt{3})=\{a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6} \mid a,b,c,d\in\mathbf{Q}\}$$ $$\mathbf{Q}(\sqrt{2}+\sqrt{3})...