3
votes
2answers
54 views

Determining final and initial object in a certain category

I am reading Paolo Aluffi's greatly entertaining book "Algebra: Chapter $0$" and I got stuck on some excercises dealing with universal properties. Let $C$ be a category, and let $A$ and $B$ be two ...
3
votes
1answer
54 views

Introductory texts in abstract algebra, and game theory taking non-standard approaches

I like to see subjects from different angles. For example in linear algebra I'm reading through Axler's text (which takes a proof based approach for math students), but I'm also checking out a text on ...
2
votes
3answers
100 views

If $G$ is a simple $f$ an homomorphism, and $A\lhd H$ is such that $[H:A]=2$, show $f(G) \subset A$

I'm stuck with the following problem. Can someone help me by providing a hint? Suppose $G$ is simple and let $f$ be an homomorphism between $G \to H$. If $\#G\ne2$, $A\lhd H$, and $[H:A]=2$. Then ...
4
votes
1answer
62 views

hint with an exercise algebra

I'm stuck with the following I hope someone could help me. Let $N$ a normal subgroup of $G$. Show that if $[G:N]=4$, exists a normal subgroup $M$ of $G$ s.t. $[G:M]=2$. My idea: Since $G/N$ has ...
1
vote
2answers
34 views

How to find a basis of an image of a linear transformation?

I apologize for asking a question though there are pretty much questions on math.stackexchange with the same title, but the answers on them are still not clear for me. I have this linear operator: ...
2
votes
5answers
131 views

Introduction and Prerequisites to Abstract Algebra

So I've seen similar questions asked, but none that really helped me out. I'm going to be a freshman in college next year, having already taken Multivariate Calculus and Elementary Linear Algebra. Of ...
0
votes
2answers
58 views

Find the galois group of the polynomial when a root is given

If $\alpha$ is a root of a polynomial $f(x)=x^3 +x^2-4x+1$ then show that $2 - 2\alpha - \alpha^2$ is also a root of $f(x)$. Use this fact to compute the Galois group of the splitting field of $f(x)$ ...
2
votes
2answers
56 views

Intersection of ideals $I=(2x)$ and $J=(2x^2)$ of $\mathbb{Z}[2x,2x^2,2x^3,\dots]$ is not finitely generated.

Consider the subring $\mathbb{Z}[2x,2x^2,2x^3,\dots]\subset \mathbb{Z}[x]$. Then show that the intersection of ideals $I=(2x)$ and $J=(2x^2)$ of $\mathbb{Z}[2x,2x^2,2x^3,\dots]$ i.e., $I\cap ...
0
votes
0answers
25 views

Proving that the $[g,x]^n=e$ if $G$ is nilpotent of degree $n$

This is an article from wikipedia which I saw wondering as to how to prove it. The question is If $G$ is nilpotent of degree $n$ then $[g,x]^n=e$ for all $x \in G$, where $[g,x]=g^{-1}x^{-1}gx$. I ...
4
votes
0answers
66 views

Analysis or (abstract) algebra first?

Which one would you recommend? I only know calculus and linear algebra when it comes to university-level mathematics. Is one required to understand the other?
7
votes
1answer
93 views

Follow up to Pinter's abstract algebra

I wanted to learn abstract algebra this summer so I bought Pinter's A book of Abstract Algebra. I was planning on reading it over the course of the summer, but just finished the last problem of its ...
3
votes
2answers
35 views

Zero divisors in ring of real valued functions.

I'm working though Pinter's A book of Abstract Algebra and would like a quick verification on a simple problem. Exercise 17.B2 asks Describe the divisors of zero in $\mathcal{F}(\mathbb{R})$. ...
0
votes
1answer
40 views

The value of to fill the gap in the proof

I have studied a paper "On Finite Groups with Given Conjugate Types I" recently. The author use many words like "obviously", "clearly", "trivial", etc. in his proof. But these "obviously" implication ...
1
vote
1answer
41 views

A question on the morphism of projective varieties

The continuation of this, my question I want to show that $X$ and $Y$ are smooth and irreducible curves then $f(X)$ is either $Y$ or a point. Note that I know the proof of this ...
2
votes
1answer
69 views

the diagonal $\Delta (Y) =\{(y,y)\in Y \times Y\}$ is closed in $\Bbb P^m \times \Bbb P^m $

Asumme that $\Pi : \Bbb P^n \times \Bbb P^m \rightarrow \Bbb P^m $is a closed map. $X\subset \Bbb P^n$ And $Y\subset P^m$ $f: X\rightarrow Y$ be a morphism of projective varieties. ...
0
votes
2answers
81 views

Every projective algebraic set can be written as the zero set of finitely many homogeneous polynomials of the same degree.

Definition: Let $I \subset k[x_0,\ldots,x_n]$ be a homogeneous ideal (or a set of homogeneous polynomials). The set $Z(I) := \{(a_0 : \cdots : a_n)\in P^n ; f(a_0,\ldots,a_n) = 0 \ \ \forall f \in ...
0
votes
2answers
31 views

Finding a cycle with a specific property

I am reading the book Dummit and Foote - Abstract Algebra . One of the exercises is to find an $n$-cycle $(n \ge 5)$, $\sigma$ such that $\sigma^k = \tau$ for some positive integer $k$, where ...
0
votes
1answer
36 views

Abstract Algebra: prove it is cyclic

I have question in referring to below link. Question. Suppose if I have [M:K]=2 and I know that K is subset of M. M:=$\mathbb{Z}_2[x]/(f(x))$ where f(x)=x$^4$+x+1. Then how this will be cyclic? I ...
0
votes
3answers
70 views

Normalizer of a subgroup of $GL_2(\mathbb{R})$

I have following subgroup of $GL_2(\mathbb{R}):$ $$A=\Bigg\{\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right),\left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} ...
0
votes
2answers
53 views

Is there any motivation for the direct sum?

I believe that everything have reason to exist . I want to learn why the direct sum exist. Is there any motivation for the direct sum to exist ?
-2
votes
1answer
175 views

The form of subrings of $k[[t]]$

I saw this question in an algebraic geometry book. I tried to solve this. But I did trivial thing, so I don't write what I did here. This is just self-studying. I want to learn how to solve. Please ...
3
votes
2answers
63 views

Galois Theory problem (primitive roots of unity)

If $e_1,e_2......,e_{p-1}$ denote primitive $p^{th}$ roots of unity. Here $p$ is prime. And set the sum of the $n^{th}$ powers of the $e_i$ as $p_n=e^n_1+e^n_2......+e^n_{p-1}$ Now I want to show that ...
1
vote
2answers
63 views

Free $A$-module isomorphic to a direct sum of copies of $A$?

Does this proposition hold even if it's not finitely generated? I think it does, since $M$ isomorphic to the direct sum of $M_i$, $M_i$ isomorphic to the direct sum of ...
0
votes
1answer
37 views

Abstract Algebra: Irreducible polynomials and automorphism

While I was studying and referring few examples I came up with few question. Hope you guys can help me solving my confusion. Irreducible polynomials problem 1 in this how to argue that why the ...
0
votes
1answer
76 views

The meaning of a symbol in the proposition

The question is what's the meaning of the symbol $\phi$? If it just a mapping,what's the mean of the equation? I guess it's $\phi(x)$, $x$ is the element of $M$. Then $\phi(x)$ is the element of ...
3
votes
1answer
66 views

Problem about Galois Extension.

$Gal(\Bbb K/\Bbb F)\cong S$. Where $\Bbb K$ is extension of $\Bbb F$. And $S= G_1\times G_2\times\cdots\times G_K$ is a solvable group. Where each $G_i$ are groups of prime power order and $o(S)= n.$ ...
1
vote
1answer
35 views

Finite field and Automorphism

Problem 1. Let S be a finite field of characteristics 2 and the map be define as $\eta$: S$\longrightarrow$S x$\longmapsto$x$^p$ Show that $\eta$ is automorphism, i.e., S is isomorphism ...
0
votes
0answers
29 views

Question of a proposition about direct product

I try to prove it's injective, surjective and homomorphism. define f(x)=(x+a1,x+a2,....,x+an),it's homomorphism. it's injective <=> the intersection of ai=0 I don't know how to prove the ...
3
votes
1answer
57 views

proposition 1.10 ii) A&M Introduction of commutative algebra

I am working through Introduction of commutative algebra and am having trouble with the following question: (I'll use f instead of the map,since I don't know how to input it.) Q1: Why there exist ...
1
vote
1answer
30 views

is there a counterexample of this map isn't surjective?

The ring A is a commutative ring with identity. I think ii) is true if they are not coprime. because for every (x+a1,....,x+an) we can find a x such that f(x)= (x+a1,....,x+an). Could you please ...
1
vote
1answer
33 views

Smallest Algebra Containing Singletons

$\Omega:=\mathbb N$. What is the smallest algebra containing all singleton $\{\omega\}$, i.e. $\{1\}, \{2\}$, and so on. Any hint, please?
4
votes
1answer
80 views

Don't understand a proposition and its proof

Proposition 5.1 from Commutative Algebra by Atiyah and Macdonald: $x∈B$ is integral over $A$,then $A[x]$ is a finitely generated $A$-module. The elements in $A[x]$ are the set of all the sum. If ...
0
votes
1answer
79 views

Abstract Algebra: Ring Homomorphism injective

Reference: Ring Homomorphism $\phi:f\to S$ is injective Referring this I have a doubt which I needed to clear. Below is my answer and query. We know $\phi:f\to S$ be ring homomorphism, where $f$ is a ...
7
votes
1answer
126 views

What's the motivation of definition of primary?

Primary ideal can be regard as the generalization of prime ideal and radical. But Why it's defined like that?It's not symmetry. Why not define like that:
2
votes
1answer
88 views

Question about some details of a proof

i) Why it's a unit can prove this proposition ii)see picture
2
votes
4answers
78 views

Are different constructions of an algebraic structure always isomorphic?

Any two complete ordered fields are isomorphic (as proved, e.g., in Spivak's Calculus; see also this question). While I understand this proof, I cannot yet appreciate why it is necessary. Given any ...
1
vote
3answers
253 views

Is it possible to learn abstract algebra no precalculus or calculus?

See above. I am trying to re-teach myself mathematics in a different manner than is formally taught (i.e., set theory, number theory, mathematical logic, abstract algebra, discrete math and then ...
0
votes
1answer
46 views

Could you please explain the detail of the proof

Proposition: Proof: Question: Why it's isomorphism?
0
votes
1answer
59 views
1
vote
2answers
56 views

primitive root of residue modulo p

I was trying to prove that for the set $\{1,2,....,p-1\}$ modulo p there are exactly $\phi(p-1)$ generators.Here p is prime.Also the operation is multiplication. My Try: So I first assumed that if ...
3
votes
5answers
256 views

Number of groups of a given order

In general, for what $n$ do there exist two groups of order $n$? How about three groups of order $n$? I know that if $n$ is prime, there only exists one group of order $n$, by Lagrange's Theorem, but ...
1
vote
1answer
71 views

How to calculate the tensor product?

This question might be stupid for you.I ask because I have no clue about it. I don't really understand what is tensor product,although I know its definition. I have search what is tensor product,so I ...
0
votes
1answer
34 views

Why these two propositions have different requirements

Proposition 2.18 is similar to 2.19. Why we need $N$ flat in 2.19? What's the difference between 2.18 and 2.19?
2
votes
1answer
78 views

Affine variety example.

I know how to show the statement. But I cannot find an example (the part I underlined by a yellow pen) please help me for finding an example. Note: For example, can I consider the following ...
2
votes
1answer
29 views

Why we cannot in general define the product of two submodules

why we can define product of two ideals of a commutative ring,but we can't in general define the product of two submodules? ...
0
votes
1answer
34 views

How to prove M/Ker(f) & Im(f) are isomomorphic

f:M->N us an A-module homomorphism. A is a commutative ring. How to prove M/Ker(f) and Im(f) are isomomorphic I can't prove this statement. But if A is a field,it's not hard to be proved. For ...
1
vote
1answer
26 views

Showing if $H,K$ are subgroups of $G$, the $\varphi(hak)=a^{-1}hak$ is bijective.

Here's my problem...I have to ultimately show that $\varphi(hak)=\varphi(hbk)\Rightarrow hak=hbk$ and I'm having issues... I am told, both $H,K$ are subgroups of $G$ and $a\in G$. The goal is to show ...
1
vote
1answer
48 views

Question about construction of an algebraic closure

$A=K[x]$,$\mathfrak{m}$ is a maximal ideal containing a principle ideal of $A$. every element of $K[x]/\mathfrak{m}$ can be described by form $f+\mathfrak{m}$. every element of $K$ is the ...
0
votes
1answer
80 views

Question about construction of an algebraic closure of a field

In Constructing algebraic closures by Keith Conrad, the author writes: Let $K$ be a field. We want to construct an algebraic closure of $K$, i.e., an algebraic extension of $K$ which is ...
0
votes
1answer
23 views

Does this opetation and structure have a name?

A is a commutative ring ,a is an ideal of A. then we can get a structure A/a called quotient ring by operation of quotient. question is :if we have a ring A/a and a set a . How to get A? This ...