3
votes
2answers
50 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
33 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
28 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
62 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
57 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
27 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
26 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
46 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
28 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 ...
0
votes
1answer
27 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
71 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
58 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
113 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
86 views

Question about some details of a proof

i) Why it's a unit can prove this proposition ii)see picture
2
votes
4answers
73 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
131 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
44 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
42 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
217 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
48 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
33 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
63 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
27 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
32 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
46 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
56 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
22 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 ...
0
votes
1answer
40 views

how to prove a=r(a)<=>a is a intersection of prime ideals.if a is an ideal≠(1) in commutative ring A.

how to prove a=r(a)<=>a is a intersection of prime ideals.if a is an ideal≠(1) in commutative ring A.r mean radical. I can't prove it,here is what I did. a is a intersection of prime ideals mean ...
0
votes
1answer
53 views

Modern Algebra: Groups

Is this the way to solve the question? Question a). Find the center of the group $S_3 \times \mathbb Z/6\mathbb Z$ Ans: $S_3$is the order of 6 element therefore, {1,(12),(23),(13),(123),(132)} and ...
2
votes
2answers
97 views

Is $m\mathbb{Z}$ not isomorphic to $n\mathbb{Z}$ when $m\neq n$?

Exercise from "Abstarct Algebra: An Introduction" by T.W.Hungerford. For each positive integer $k,$ let $k\mathbb{Z}$ denote the ring of all integer multiples of $k$. Prove that if $m\neq n$, then ...
1
vote
1answer
52 views

Question about some details of a proof of Chinese Remainder Theorem

In the proof of 3rd proposition I can prove the intersection of all ideals is the kernel of the map, but why does it imply this proposition is true?
1
vote
1answer
33 views

A question about a detail of proof

proposition: x∈The Jacobson radical <=> 1-xy is a unit in commutative ring A for all y∈A I have proved (=>) I don't figure out a detail of the proof of (<=). Here is the proof on book: ...
0
votes
3answers
60 views

How to prove a simple proposition about local rings and maximal ideals

(The word ring shall mean a commutative ring with an identity element in this question.) Actually, there is a proof about this proposition, but I don't get it, even the first step. Proposition: ...
0
votes
1answer
45 views

How to prove m is maximal iff A/m is a field?

m is a maximal ideal of a commutative ring A. then m is maximal iff A/m is a field. Use Lattice theorem we get there is a bijection between m and an ideal of A/M. A/M is a field =>the only odeals in ...
1
vote
1answer
34 views

How to prove there exists a bijection between the ideals of $A/a$ and the ideals of $A$ containing $a$

original proposition is there is a one-to-one order-presserving correspondence between the ideals of $A$ which contain $a$ and the ideals of $A/a$. I think one-to-one correspondence mean ...
2
votes
2answers
56 views

How to prove “ideal $I$ is prime iff $A/I$ is a integral domain ”?

$A$ is a commutative ring with identity. $I$ is a ideal of $A$. then ideal $I$ is prime iff $A/I$ is a integral domain. here is what I thought $(\Rightarrow)$ We want to prove $A/I$ is a integral ...
0
votes
2answers
60 views

Question about some algebra theorem

1. order-presserving = monotonic But we haven't define order structure on the ring. 2. I try to prove x is a unit <=> (x)=A ,and fail. That's what I think: => we want to prove (x)=A. ...
0
votes
1answer
34 views

A proof problem about congruence relation

Given a ring R and a two-sided ideal I in R, we may define an equivalence relation ~ on R as follows: a ~ b if and only if a − b is in I. Using the ideal properties, it is not difficult to check ...
7
votes
0answers
201 views

Modern research into Grassman's “theory of forms”?

I quote from Petsche's Hermann Graßmann: Biography (emphasis mine): The mathematical part of the book begins with the conception of the “General Theory of Forms”. Starting with a perspective on ...
1
vote
0answers
36 views

Resources for self-learning “relational” abstract algebra? [please see body of post for details]

I have been studying Grassman and Clifford algebras a bit, and it is fascinating to see how, for example, the rules defining the inner product operator are enough to the capture something of the ...
3
votes
1answer
224 views

Prerequisites for studying Homological Algebra

I have read the answers here and here and need to ask something more. I wish to study the book on Homological Algebra by Weibel but am not sure of the prerequisites. In particular how much ...
0
votes
1answer
27 views

Field extension $F\subseteq L_1$ and $F\subseteq L_2$ and $[L_1L_2:F]<[L_1:F][L_2:F]$.

I'm searching for an example of field extensions $L1$, $L2$ of $F$ for which $[L_1L_2:F]<[L_1:F][L_2:F]$. Infact I'm trying prove the problem below. So any hint can be helpful. Let $K$ be a ...
1
vote
1answer
49 views

$\pi^{b}$ transcendental?

I asked a similar question in a different thread but that one got answered and I thought this would be the natural extension. Is $\pi^{b}$ transcendental for any algebraic $b$? Is this a known ...
2
votes
1answer
102 views

Transcendence of $\sqrt{\pi}$

So it is known that $\pi$ is transcendental. With a little thought I was able to prove that $k\pi$ and $\pi^{k}$ for all $k\in\mathbb{Z}$ was transcendental. After that I thought about $\pi^{b}$ for ...
3
votes
2answers
54 views

Condition such that $\langle{S}\rangle=S$

So let $S\subseteq{G}$ where $G$ is a group and $S$ an arbitrary subset. Let $\langle{S}\rangle$ be the subgroup in $G$ generated by $S$. What is the condition such that $\langle{S}\rangle=S$? My ...
9
votes
2answers
596 views

Learning Abstract Algebra for a graduate degree

I would like to do a graduate degree in mathematics, and I have a full year before I will be able to do so (for personal reasons). I mainly have my weekends available to study. I am interested in ...
0
votes
1answer
83 views

Abelian Group elements and inverses

Let G be a finite abelian group, say, $G={e,a_1,a_2...a_n}$ Prove the following: a)$(a_1a_2...a_n)^2=e$ b)If there is no element x $\neq$ e, x=x^(-1), then $a_1a_2...a_n=e$ c)If there is exactly ...
7
votes
4answers
248 views

Advanced Mathematics

I am a high school student and would like to pursue a career in mathematics and I am hoping to find a serious explanatory book on math (geometry, algebra, calculus, functions and trigonometry) for ...