0
votes
1answer
35 views

Proves or counterexamples in retraction and coretractions of modules

Any tip for proving or counterexamples that the following morphism of $\mathbb Z$-modules ${\mathbb Z} \to {\mathbb Q}$ is not a retraction and ${\mathbb Q} \to {\mathbb Q/\mathbb Z} $ is not a ...
1
vote
1answer
63 views

Irreducibility of $\frac{X^{p^n}-1}{X^{p^{n-1}}-1}$.

Exercise: Let $p$ be a prime number. Then, the polynomial \begin{equation} \frac{X^{p^n}-1}{X^{p^{n-1}}-1} \end{equation} is irreducible over $\mathbb Z[X]$, for any integer $n \geq 1$. I'm able to ...
0
votes
0answers
16 views

Finite p-primary group question

Problem is rotman's Intrdouction to theory of group, Exercise 6.8. Let G be a direct sum of b cyclic groups of order $p^{m}$. If $n< m$, then $p^{n}G/p^{n+1}G $is elementary and ...
5
votes
0answers
161 views

Elementary proof of irreducibility criterion

From ``Problems from the Book'' by Andreescu and Dospinescu, the following irreducibility criterion is presented: Let $f$ be a monic polynomial with integer coefficients and let $p$ be a prime. If ...
3
votes
2answers
111 views

Multiple choice question regarding the ideal of $C[0, 1]$

Let $C[0,\ 1]$ be the ring of continuous real-valued functions on $[0,\ 1]$, with addition and multiplication defined pointwise. For any subset $S$ of $C[0,\ 1]$ let $$Z(S) =\{f \in C[0,\ 1] \mid f(x) ...
1
vote
1answer
93 views

Problem 9.7 - Lie Algebras - Humphreys

Let $\alpha,\beta\in\Phi$ span a subspace $E'$ of $E$. Prove that $E'\cap\Phi$ is a root system in $E'$. Prove similarly that $\Phi\cap(\mathbb{Z}\alpha+\mathbb{Z}\beta)$ is a root system in E' (must ...
4
votes
3answers
113 views

One-to-one homomorphism $f: F\to F$ which is not onto?

The following question comes from a past qualifying exam: What is an example of a ring homomorphism $f: F\to F$ such that $f$ is one-to-one but not onto? (Here $F$ is assumed to be a field.) ...
1
vote
1answer
55 views

Indecomposable module over a PID, Jacobson's Basic Algebra I - followup

I am trying to solve the question which already appeared here: Indecomposable $D$-module $M$, Jacobson's Basic Algebra I. Let $D$ be a PID. Let $M$ be a finitely generated $D$-module. Then $$ M\ ...
2
votes
1answer
58 views

Equalities in Groups, How prove this? [duplicate]

Let $ H,K≤ G$, for all $g ∈ G $ , $$ \frac {|H||K|}{|H∩(gKg^{-1})|}= \frac {|H||K|}{|(g^{-1}Hg)∩ K|} $$ I try to show this but I do Know how to attack this exercise.
2
votes
1answer
1k views

Solution Manual for Chapters 13 and 14, Dummit & Foote

I bought the third edition of "Abstract Algebra" by Dummit and Foote. In my opinion this is the best "algebra book" that has been written. I found several solution manual but none has solutions for ...
2
votes
1answer
74 views

Indecomposable $D$-module $M$, Jacobson's Basic Algebra I.

I can't do the left direction of this problem (this is the exercise 3 in the section 3.9) Let $M$ be a finitely generated $D$-module over a p.i.d. Show that $$ M\ \text{is ...
1
vote
1answer
94 views

Determining the structure of the $\mathbb{Z}$-module $\mathbb{Z}^3/K$, with $K=\langle (2,1,-3),(1,-1,2)\rangle$

Well, this is the exercise: Determine the structure of $\mathbb{Z}^{(3)}/K$ where $K$ is generated by $f_1=(2,1,-3)$, $f_2=(1,-1,2)$. Looking at the proof of the fundamental structure theorem ...
2
votes
2answers
261 views

Exercise in Jacobson's $Basic\ Algebra\ I$, Chapter 3

Well, I even don't understand the problem. Let $R$ be a ring and let $(e_1,\ldots,e_n)$ be a base for $R^{(n)}$. If I define: $$ f_j=\sum_{j'=1}^n a_{jj'}e_{j'} $$ for all $j \in ...
1
vote
1answer
77 views

$\phi \times \theta$ is faithful iff $(|Z(H)| ,|Z(K)|)=1$ for faithful characters $\phi \in Irr(H)$ and $\theta \in Irr(K)$ .

Let $G = H \times K$. Let $\phi \in \operatorname{Irr}(H)$ and $\theta \in \operatorname{Irr}(K)$ be faithful. Show that $\phi \times \theta$ is faithful iff $(|Z(H)| ,|Z(K)|)=1$. Problem 4.3 of ...
1
vote
1answer
73 views

Two problems in ideal and radical

Let $R$ be a commutative ring with multiplicative identity. Let $I$ be an ideal of $R$. Let $S=\{r \in R: r^n \in I\mbox{ for some natural number }n\}$. Show that $S$ is an ideal of $R$. Give an ...
5
votes
1answer
456 views

Simple module is isomorphic to R/M where M is a maximal ideal

In Michael Artin's Algebra textbook page 484 Chapter 12 Exercise 1.6: A module is called simple if it is not the zero module and if it has no proper submodule. (a) Prove that any simple module is ...
2
votes
3answers
736 views

Homework - Prove that a given set is a group [duplicate]

I have a homework question and I don't know how to approach this exercise. The exercise is the following: Let's suppose $G$ is a set with binary function * defined for its members, which is: ...
2
votes
1answer
272 views

Question on groups of order $pq$

Let $G$ be a group of order $pq$, $p>q$ and $p$, $q$ are primes. Then prove that If $q\mid p-1$ then there exists a non abelian group of order $pq$. Any two non-abelian groups of ...
10
votes
1answer
276 views

Question on a homomorphism of a set G.

I'm having difficulty showing the given a map, say $\phi(z)=z^k$, is surjective. This question is from D & F section 1.6 - #19 Let $G$ =$\{z \in \mathbb C|z^n=1 \text{ for some } n \in \mathbb ...
5
votes
1answer
144 views

Finding an algebra of smooth functions on a manifold with a given product.

I am having trouble with the following exercise from Jet Nestruev's Smooth Manifolds and Observables. It is one of the first exercises in the book and I have no idea how to approach this type of ...
3
votes
2answers
139 views

Nice exercises on resultants

I would like to ask if some one knows a source (a book, or lecture notes ect) that contains several nice exercises on resultants of polynomials (it would be nice if there were some solutions as well ...
3
votes
1answer
428 views

Book of problems in abstract algebra [duplicate]

Possible Duplicate: Good problem book on Abstract Algebra What are some books similar to Problems in Real Analysis: Advanced Calculus on the Real Axis by Radulescu, Radulescu and Adreescu ...
7
votes
3answers
2k views

Good problem book on Abstract Algebra

I am currently self-studying abstract algebra from Artin. In that background, I am looking for a problem book in a spirit somewhat similar to Problems in Mathematical Analysis by AMS so that I have a ...