Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, and other advanced topics.

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Examples of rings with idempotent elements

As a part of my studies in ring theory, I've encountered the concept of an idempotent element, i.e., an element $e$ such that $e^2=e$. Are there some interesting examples of rings with idempotent ...
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1answer
130 views

Field of sets versus a field as an algebraic structure

During my studies of real analysis I've come across the definition of a field (or algebra) of sets. My question is: What is the connection between this structure and the structure of a field or ...
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1answer
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$|\{ x\in X: g.x=x \space\space\space \forall g\in G \}| = |X|\space mod \space p$

Let $G$ be a p-group. $|G|=p^n$ for some n. Let X be a finite set so that $\,p\nmid |X|\,$, G acts upon X. Denote $A:= \{ x\in X: g.x=x \space\space\space \forall g\in G \}$ I am trying to show ...
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3answers
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Why are the surreals considered “recreational” mathematics?

One of my college math professors once remarked to me that it was interesting that John Conway's two "biggest" contributions to math were both recreational: the Game of Life and the Surreals. No one, ...
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0answers
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Gröbner Basis and linear basis

Let $I$ be an ideal of a polynomial algebra $A$ with a Gröbner basis $G$. Suppose we know how to describe the leading terms of all elements in $G$, denoted by $\{i_1,\dots,i_k\}$, so that we can give ...
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3answers
59 views

Group of order $p^2$ is commutative with prime $p$

Please help me on this one: Let $p$ be a prime number, show that each group of order $p^2$ is commutative. If you do not mind at all, could you please not give me the elegant explanation, but ...
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2answers
37 views

every ideal is contained in a maximal ideal

The statement is: In a commutative ring with 1, every proper ideal is contained in a maximal ideal. and we prove it using Zorn's lemma, that is, $I$ is an ideal, $P=\{I\subset A\mid A\text{ is ...
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1answer
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inverse limit in the plane

What stuff can I say about inverse limits regarding the mapping of $[0, 1]$ onto $[0,1]$ given by $$f(x) = \left\{ \begin{array}{ll} 2x & \mbox{if } 0 \le x \le {1\over2}\\ 1 & \mbox{if ...
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0answers
52 views

Which convex $2n$-gons have symmetry group $D_n$ instead of $D_{2n}$?

The equilateral octagon $M$ in the first image has the same symmetry group as the small embedded square - namely the dihedral group $D_4$ - with $8$ elements and generators ${x,y}$ with $x^4 = e, y^2 ...
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1answer
36 views

Are the two ways of creating an $S^{-1}A$ algebra equivalent?

Let $f:A\to B$ be a ring homomorphism and $S$ be a multiplicative set, define $S^{-1}B$ to be $B\times S$ with equivalence relation $(b,s)\sim(b',s')$ iff $\exists t\in S$ such that $t(sb'-s'b) = 0$. ...
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2answers
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any $2$-dimensional rep of a finite, non-abelian simple group is trivial

Let $G$ be a finite, non-abelian simple group. How would I go about proving that any $2$-dimensional representation of $G$ is trivial? If it helps, I know how to do it when we're considering ...
277
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7answers
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“The Egg:” Bizarre behavior of the roots of a family of polynomials.

In this MO post, I ran into the following family of polynomials: $$f_n(x)=\sum_{m=0}^{n}\prod_{k=0}^{m-1}\frac{x^n-x^k}{x^m-x^k}.$$ In the context of the post, $x$ was a prime number, and $f_n(x)$ ...
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0answers
28 views

MCS meet all prime ideals

let A be a commutative ring, is there any multiplicatively closed subset S (not containning 0), s.t. every prime ideal in A intercept S is not empty? My thinking is that there is 1-1 ...
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2answers
34 views

What are the notations $k^{\prime n}$ and $\varphi^n$ in algebra?

I would like to understand what the following problem says: Let $k$ be a commutative ring and $f\in k[X_1,\ldots,X_n]$. Let $k^\prime,k^{\prime\prime}$ be commutative $k$-algebras and ...
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1answer
38 views

Define a projection homomorphism and find the kernel

I was given the projection homomorphism $\mathbb{Z}_4 \times \mathbb{Z}_3 \to \mathbb{Z}_3$ and asked to find it and come up with the kernel. I came up with $\phi(x,y)= x$ such that $x \in ...
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0answers
43 views

Irreducible polynomial and primes [duplicate]

Let $n$ be a prime number. How can I show that the polynomial $f(x) = x^{n-1} + x^{n-2} + x^{n-3}+ \cdots + x+ 1$ is irreducible over any finite field?
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2answers
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Why isn't a (noncommutative) ring with only trivial ideals a division ring?

We know in a commutative ring, if the only ideals are trivial and the whole ring, then the ring is a field, which is proved by every ideal is contained in a maximal ideal, which is proved by Zorn's ...
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4answers
421 views

This ideal is prime or not?

I'm trying to prove this ideal $$I=(x^2+y^2+x,x+y+xy)\subset \mathbb C[x,y]$$ is prime. I supposed that $I$ is prime and I'm using the classical method to prove $I$ is prime: If $ab\in I$, ...
6
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2answers
298 views

Origin of the modern definition of the tensor product

Due to whom is the modern (i.e. via its universal property) definition of the tensor product, and in which article was it communicated?
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3answers
38 views

orthogonal and special orthogonal group of dimension $2$, group of isometries of $S_1$, $\mathbb{R}^2$ [on hold]

In my abstract algebra class, my teacher gave us this problem as to help review for the final. Unfortunately, I am not very well versed with linear algebra so I don't understand all that well what ...
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1answer
60 views

Quotient of polynomial ring in two variables is a PID

With $K$ a field and $K[x,y]$ the polynomial ring over it in two variables, the quotient ring of it over the ideal generated by $1-xy$ is a PID. I've tried using Noetherianess but haven't gotten ...
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2answers
33 views

Group Transitive Action's Effect on Stabilizers's Conjugacy

I am looking for guidance for two problems on group action, one of them is here and the other one has just been posted earlier: Assume that $G$ operates on a set $\Omega.$ Show that, if $G$ acts ...
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4answers
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Difference between Ring and Algebra?

In mathematics, I want to know what is indeed the difference between a ring and an algebra? Thanks!
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3answers
90 views

Why not define $|v| = -1$? [on hold]

I was wondering why if we have $i^2 = -1$, why not have a "number" $v$ such that $|v| = -1$? Does anything interesting arise from considering this system? The only thing I could come up with was: ...
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5answers
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Is $\mathbb{Z}[x]$ a principal ideal domain?

Is $ \mathbb{Z}[x] $ a principal ideal domain? Since the standard definition of principal ideal domain is quite difficult to use. Could you give me some equivalent conditions on whether a ring is a ...
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2answers
45 views

Minimal subring of complex numbers

Let $\alpha$ be a root of $X^3+X^2-2X+8$. My question is if $\mathbb Z\left[\alpha,\frac{\alpha+\alpha^2}{2}\right]=\{a+b\alpha +c\frac{\alpha+\alpha^2}{2}:a,b,c\in\mathbb Z\}$? Thank you all.
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1answer
64 views

Why $ \mathbb{Z}[x]$ is not Principal Ideal Domain [duplicate]

$ \mathbf{Z}[x]$ is not PID. we know $\mathbb Z$ is a Unique Factorization Domain, so $\mathbb Z[x]$ is UFD, but why isn't it PID (since I think $\mathbb Z$ is PID)?
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1answer
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A question on Weyl algebra $A_1$

This question is from A Primer of Algebraic D-Modules by S.C. Coutinho, Ex 2.4.10. In the following $A_1$ means the Weyl algebra generated by $x$ and $d$. Let $f: A_1^2\to A_1$ be the map defined ...
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3answers
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Intersect of Stabilizers is a Normal Group

I am looking for guidance for two problems on group action, one of them is here and the other one will be posted in another page: Assume that $G$ operates on a set $\Omega.$ Show that ...
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1answer
334 views

Theorems similar to Euler's theorem ($a$, $n$ are not coprime)

It is well known that if $\gcd(a,n)=1$, then $a^{ϕ(n)}=1$ mod $n$. Are there any results similar to Euler's theorem that can be used when $a$ and $n$ are not coprime. Feel free to add any ...
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3answers
111 views

how to show $\mathbf{Q} $ is not free

we know torsion free plus finitely generated $\rightarrow$ free and that $\mathbf{Q}$ is torsion free is easy. But how to show Q is not finitely generated? and not free?
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2answers
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Dihedral Group Computations

Let $n > 1$ be an integer and let $\theta = \dfrac{2\pi}{n}$. Let $P$ be the regular $n$-gon with vertices ($\cos i\theta$, $\sin i \theta$) for $i \in \mathbb Z_n$. The dihedral group $D_n$ is the ...
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0answers
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$\underset{\longrightarrow}{\lim}\operatorname{Hom}(R^n, M_\lambda) \cong \operatorname{Hom}(R^n, \underset{\longrightarrow}{\lim}M_{\lambda})$

Let $R$ be a ring, $\Lambda$ a category of $R$-modules $M_{\lambda}$. Prove the colimit $$ \underset{\longrightarrow}{\lim}\operatorname{Hom}(R^n, M_\lambda) \cong \operatorname{Hom}(R^n, ...
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1answer
105 views

If $G/G'$ is finite, then $|Z(G)| < \infty$

Let $G$ be an infinite group. Suppose that $G/G^{\prime}$ is a finite group, where $$G^{\prime}= \left\langle xyx^{-1}y^{-1}\ \middle\vert\ x,y \in G\right\rangle.$$ Prove that $|Z(G)| < \infty$.
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1answer
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Finite fields and arithmetic

For every prime number $p$ and every positive integer $k$, there is a field with exactly $p^k$ elements. When $k=1$, it's just the integers$\bmod p^k$, and when $k>1$, it's not. So if I want the ...
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2answers
28 views

Is there any automorphism $f$ that satisfies these requirements? [on hold]

Suppose that $\Bbb R $ is the set of real numbers. Is there any automorphism $f$ from $(\mathbb R,+)$ to $(\mathbb R,+)$ of the following form? $$f(x)=kx, \quad k \neq 0,1 , \quad k \in \Bbb R$$
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2answers
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Number of subgroup of order $p^2$ in $\mathbb{Z}_{p^{2}} \times \mathbb{Z}_{p^{2}}$

Let $G = \mathbb{Z}_{p^{2}} \times \mathbb{Z}_{p^{2}}$. How many subgroups does $G$ has of order $p^2$? I know there are only 2 cases of the subgroup H, H can be isomorphic to $Z_{p^{2}}$ or $ ...
4
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1answer
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$M$ noetherian, $f$ endomorphism of $M$, $\operatorname{coker}f$ has finite length, then $\operatorname{coker}f^n$ and $\ker f^n$ have finite length.

Let $M$ be noetherian and let $f$ be an endomorphism of $M$. Suppose that $\operatorname{coker}f$ has finite length. Prove that both $\operatorname{coker}f^n$ and $\ker f^n$ have finite length ...
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0answers
33 views

Use Pohlig-Hellman to solve discrete log

We have $$7^x = 166 \pmod{433}$$ I need to find $x$ using the Pohlig-Hellman algorithm.
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1answer
49 views

Number of group actions [on hold]

In how many ways can the group $\mathbb{Z}_5$ act on $\{1,2,3,4,5\}$.
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3answers
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Injective Ring Homomorphism

I seem to be having the wrong impression of what $p$ stands for; is $p(x)=x(x+1)(x+2)$ or is it something else? Clarification would be appreciated so that I can complete the lemma below. Consider ...
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1answer
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How to find fast modular exponentiation?

I need to find $$5448^{5^3} \pmod{11251}$$ and $$6909^{5^3} \pmod{11251}$$ fast? I couldn't calculate it with normal tricks.
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0answers
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Density of odd vs even group orders that are not forced to be simple by Sylow's Theorem

In Dummit & Foote's Abstract Algebra text, I've just solved the following two Exercises: Write a computer program which (i) gives each odd number $n<10,000$ that is not a power of a ...
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1answer
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Order of elements in finite fields

Show that 5448 has order 5^4 in Z/11251Z. How do I show this quickly, I know that all elements must have an order that divides order(Z/11251) according to lagrange's theorem but how do i pinpoint that ...
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1answer
20 views

Does the set operation define a binary operation on G?

Consider the set G = {0,{1},{2},{1,2}}. Does the set operation intersection de fine a binary operation on G? Does the set operation union de fine a binary operation on G? Is < G,(union) > a group? ...
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2answers
41 views

Show that $f _ a $ is a Homomorphism

For a fixed element $a$ is a group $G$, define $$f _ a (x) = a ^ {−1} xa , x \in G$$ Show that $f _ a $ is a Homomorphism. I know that to show that a mapping $f:G \rightarrow G'$, Where $G$ and $G'$ ...
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1answer
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NTRU cryptosystem

For the NTRU cryptosystem (as described here http://en.wikipedia.org/wiki/NTRUEncrypt), why is it really easy for Eve to decrypt if $p$ divides $q$. My answer was that when Eve sees $e(x)= ...
5
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4answers
109 views

Prove that $\mathbb{Z}$ and $\mathbb{Q}$ are not isomorphic as groups

I'm working on the following problem: Prove that $\mathbb{Z}$ and $\mathbb{Q}$ are not isomorphic as groups. Here is my attempt at a solution: If $\mathbb{Z} \cong \mathbb{Q}$, then there must ...
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1answer
84 views

Is $(G,*)$ commutative? [on hold]

$(G,*)$ is a group and for some three consecutive integers $i=j,j+1,j+2$, it satisfies $(a*b)^i=a^i*b^i$ for every $a,b\in G$. Is $(G,*)$ commutative?
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2answers
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List the elements of G

So we are asked to write out the elements of G and H where G= $ \mathbb{Z}/ <20> $ and H = $<4, 20>$ . I understand how to do H and I got: {$0 + <20>, 4 + <20> , 8 + ...