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.

learn more… | top users | synonyms (1)

1
vote
0answers
9 views

Can this be solved for f(n) =?

While working upon a partial sum formula for the harmonics, I came across a necessity for the function defined below f(n(n-1)/2 +1) = n! Can it be solved for f(n) =?
1
vote
1answer
10 views

Group under composition

Which proper subsets of $S_3$ for a group under composition? I'm not really sure how to approach this. I know the four requirements of groups - identity, closure, inverse and associativity. And I ...
5
votes
0answers
19 views

Abelian groups whose automorphism group is a $p$ group

$\def\Aut{\operatorname{Aut}}$ Let $G$ be a finite abelian group such that $\Aut(G)$ is an $p$ group ,that is, $|\Aut(G)|=p^n$ . Then can we determine the cyclic decomposition of $G$ or at least the ...
0
votes
0answers
12 views

What is the order of the tangent of $C_F$ at the point $P$?

In my lecture notes there is the following about inflection points: Definition: A point $P=[x, y, z]$ of an algebraic curve $C_F=V(F)$ is called inflection point of $C_F$ if $P$ is not a singular ...
4
votes
0answers
46 views

An example of a nilpotent group

Is there an example of a nilpotent group such that $G/G'$ is (non-trivial) torsion-free while $G$ is not? I cannot think of any example of this kind and I think that it is not proved any result like ...
0
votes
0answers
7 views

Bezout's Theorem-intersection multiplicity

Bezout's Theorem: $K$ algebraic closed Let $X=V(F),y=V(G)$ two projective curves of $\mathbb{P}^2(K)$ with degree $m$ and $n$ respectively that do not have a common component. Then: ...
1
vote
1answer
22 views

Proving that the char(R) is non-zero.

Let $R$ be an integral domain and assume that for some non-zero $a \in R$, that $\exists n_a \in \mathbb{N}$ such that $n_a a = 0$. Prove that $R$ has non-zero characteristic. So here is my thinking ...
1
vote
0answers
20 views

I need example to satisfy in this theorem (Hall Subgroup)

I need example to satisfy in this theorem: let $H$ be a subgroup of $G$ such that $\mid G : H \mid$ is a $\Pi$-number.If there is a nilpotent subgroup $K$ of $G$ such that $G=HK$ then $G=HK_{\Pi}$, ...
1
vote
1answer
11 views

Bezout-If two curves intersect at $m\cdot n$ points then the intersection multiplicity is $1$

In my lecture notes, after the Bezout theorem there is the following collary: If the plane projective curves, $x=V(F), y=V(G)$, intersect at exactly $m \cdot n$ discrete points, then the ...
0
votes
0answers
19 views

$\mathbb{Z}[\zeta_n]$ is a PID for $n=3,4,5$ using Minkowski theory

I want to show that $\mathbb{Z}[\zeta_n]$ is a PID for $n=3,4,5$ using Minkowski theory. I know that if the class group is trivial, then it is a PID. Is this helpful to show the claim or how else can ...
0
votes
2answers
39 views

Geometric Interpretation of S3

My impression was that the symmetric group $S_3$ acts on the vertices of a labeled triangle. However, I am not sure this is the case anymore, because of the following. (The triangle is labeled as ...
0
votes
0answers
13 views

Projective algebraic curves-affine curves

At the projective algebraic curves there are similar identities to affine curves. Intersection points of projective algebraic curves. The meanings order of point of the curve $F$ intersection ...
1
vote
0answers
17 views

Referencing the construction of a left loop

We call left loop a magma $(L,\cdot)$ such that for all $(a,b)\in L\times L$, exists only one $x\in L$ such that $a\cdot x=b$, exists one $e\in L$ such that $e\cdot x=x=x\cdot e$ for all $x\in L$. ...
0
votes
2answers
36 views

Ring functions which 'respects' homomorphisms

Let $R$ be a ring, and $\varphi \in End(R) $ an homomorphism. We define $$ S_{\varphi} = \{f:R\ \rightarrow\ R \ | \varphi(a) = \varphi(b) \ \Rightarrow \ \varphi(f(a))=\varphi(f(b))\ \} $$ In a ...
2
votes
2answers
49 views

Prove or disprove that Q[√2] is a field

Let $R = \mathbb{Q}[\sqrt2]$ = {$\alpha + \beta\sqrt2 \;|\; \alpha, \beta \in \mathbb{Q}$} and $z = a + b\sqrt2$ with $a,b \in \mathbb{Q}$ Prove or disprove R is a field. Prove $(a + b\sqrt2)(a - ...
3
votes
0answers
23 views

Calculate the singular points of affine curve

I want to calculate the singular points of the affine curve $$f(X,Y)=(1+X^2)^2-XY^2 \in \mathbb{C}[X,Y]$$ The point $P=(x,y)$ is singular $\Leftrightarrow$ If $x=0$ we find $y=0$ and then from the ...
3
votes
1answer
27 views

Sylow subgroup of a group

Let $G$ be a group such that $\vert G\vert=231$. I have to show that the unique Sylow 11-subgroup of $G$ is contained in the center of $G$ I proceed as follows: Since the number of Sylow 11-subgroup ...
0
votes
1answer
17 views

Find the irreducible polynomial of $\zeta_{12}$ and $\zeta_9$ over the field $Q(\zeta_3)$.

Let $\zeta_n=exp(2 \pi i/n)$ Find the irreducible polynomial of $\zeta_{12}$ and $\zeta_9$ over the field $Q(\zeta_3)$. Here I can find polynomial satisfied by these elements over $Q(\zeta_3)$ but i ...
1
vote
3answers
56 views

When $\Bbb Z_n$ is a domain. Counterexample to $ab \equiv 0 \Rightarrow a\equiv 0$ or $b\equiv 0\pmod n$

Suppose $$ab \equiv 0 \mod n$$ and that $a$ and $b$ are positive integers both less than $ n$. Does it follow that either $a | n$ or $b | n$? If it does follow, give a proof. If it doesn’t, then ...
2
votes
0answers
14 views

$V$ is irreducible exactly then when $I(V )$ is a prime ideal

If $V$ is an algebraic set of $K^n$, show that $V$ is irreducible exactly then when $I(V )$ is a prime ideal of $K[X_1, X_2, \dots, X_n]$ . Let $V$ be irreducible. We suppose that $I(V)$ is not a ...
0
votes
0answers
15 views

Why is abelian group structure needed in the definition of a ring? [duplicate]

Why do we require $(R, +)$ to be an abelian group for $(R, +, \cdot)$ to be a ring? Why don't we study $(S, +, \cdot)$ where $(S, +)$ is a group and $(S, \cdot)$ is a semigroup and distributivity ...
0
votes
1answer
24 views

Is it possible that a finitely generated ring has an ideal that is not finitely generated

Sorry if this is duplicated. I couldn't find an exact answer of my question. One definition of Noetherian ring is: A ring $R$ is Noetherian if all its ideals are finitely generated. I know there are ...
1
vote
1answer
15 views

Additive Order and Ring

In an integral Domain, Additive order of each element is same and prime.(each equal to characteristic of ring) Is converse also true? Q1. Given additive order of each element same in a ring, would it ...
0
votes
0answers
20 views

group hyper_(Abelien_by_finite)groups has non trivial normal subgroup H of G such that H finite or Abelien [on hold]

Let G be group hyper_(Abelien_by_finite)group, show that G has non_trivial normal subgroup H of G such that H finite or Abelien. hyper_(Abelien_by_finite)groups by definition if it has an ascending ...
2
votes
2answers
104 views

Ring with each element finite order but not of finite characteristic

What is example of ring without identity with each element of finite additive order but not of finite characteristic. Motivation: A ring with identity and having identity of finite additive order is ...
6
votes
0answers
52 views

How do I prove that $S_A\cong S_B\implies |A|=|B|$?

Let $A,B$ be infinite sets such that $S_A\cong S_B$. (Symmetric groups are group isomorphic) How do I prove that $|A|=|B|$? The only proof I know uses Axiom of choice. (That is, using AC to give ...
1
vote
2answers
22 views

Sign of Composition of Permutations

Let $\sigma$ be a permutation in $S_5$ with a sign of $-1$. Let $\pi$ be any other arbitrary permutation of $\{1,...,5\}$. What is the sign of the composition $\pi^{-1} \sigma \pi$? Is there any ...
1
vote
1answer
22 views

Proving the minimality of an element order

Assume that I have a finite group G of order n with a generator g, and also assume that I want to prove that $\frac{n}{gcd(n,m)} $ is the order of an element $x = g^m \in G$. First , I showed that ...
2
votes
1answer
29 views

Intuition of coset of a subgroup

Hey guys I am trying to form the intuition that distinct left coset of subgroups are actually disjoint. I understand the proof constructed but I don't think I get the intuition behind why that the ...
1
vote
3answers
59 views

Set $A$ not closed under $\star$ then $A$ not a group under $\star$?

I am currently doing some exercises. I have been through some examples of solutions in other books that questioned me. I know well that $(A,\star)$ is a group if it satisfies the following points, ...
1
vote
2answers
32 views

Invertible element of $S$

Let $S=\mathbb{Z}[\sqrt{2}]$ = {$a+b\sqrt2|a,b\in \mathbb{Z}$} and $R = \mathbb{Q}[\sqrt2]$ = {$\alpha + \beta\sqrt2 | \alpha, \beta \in \mathbb{Q}$}. Consider $x=3+2\sqrt2$ and $y = 3+4\sqrt2$ ...
1
vote
1answer
26 views

In a Category, Is the Set of Morphisms Between Objects Defined to Be All Possible Morphisms?

For instance, if I have a category $\mathfrak{M}$ whose objects are families of morphisms $\{f_i\colon A_i\to B\}_{i\in{I}}$, then if we consider two such objects, say $C=\{f_i\colon A_i\to B\}$ and ...
1
vote
0answers
66 views

Can we use the Nullstellensatz?

In $\mathbb{C}[x, y, z]$ we have that $V=\{y-x^2, z-x^3)=\{(t, t^2, t^3) | t \in \mathbb{C}\}$. To show that $$I(V(y-x^2, z-x^3))=\langle y-x^2, z-x^3\rangle $$ can we use the Nullstellensatz?? ...
4
votes
1answer
23 views

Find $f_{1}, f_{2}$ in $\mathbb{Z_{6}}[x]$ such that deg$(f_{1})$ = deg$(f_{2}) = 2$ and deg$(f_{1}+f_{2})=1$

Find $f_{1}, f_{2}$ in $\mathbb{Z_{6}}[x]$ such that deg$(f_{1})$ = deg$(f_{2}) = 2$ and deg$(f_{1}+f_{2})=1$ Find $g_{1}, g_{2}$ in $\mathbb{Z_{6}}[x]$ such that deg$(g_{1})$ = deg$(g_{2}) = 1$ and ...
2
votes
1answer
31 views

Highest Common Factor

Which is the smallest integer number $a$ so that the highest common factor of $a$ and $a+5$ is not $1$ and the highest common factor of $a$ and $a+7$ is not $1$ either? I think that it is $35$. Am I ...
3
votes
2answers
41 views

Polynomial ring addition in $\mathbb{Z_{6}}$

I know this is a very simplest question ever. But, I need help with understanding it. So here it goes... Let, $f(x) = \bar{1}+\bar{2}x+\bar{3}x^2$ and $g(x) = \bar{4}+\bar{5}x$ $\in \mathbb{Z_{6}}.$ ...
1
vote
1answer
27 views

If $G/Z(G)$ is of size $qp$ and $p-1$ is not divisible by $q$ then $G/Z(G)$ is cyclic?

I have $G/Z(G)$ with size $pq$, $p, q$ are prime and $p>q$; $(p-1) $ is not divisible by $q$ How do I deduce from the above that $G/Z(G)$ is cyclic?
1
vote
1answer
46 views

Which algebraic intuition can be used in fields

I wonder what basic laws of arithmetic of reals e.g. $x^n y^m = (xy)^{m+n}$ holds for fields. Every time I take some book on abstract algebra it proves very abstract and unpractical properties. So I ...
0
votes
0answers
35 views

Group Action and “nice” Approximation!

Studying the action of a group $G$ on a set $X$ is naturally the same as looking at the group homomorphism $\alpha: G \rightarrow Perm(X)$. So, for a given group $G$, classifying all sets $X$, on ...
2
votes
1answer
29 views

A question about $rad(M)$

We know that for any $R$-module $M$ the radical $rad(M)$ is the sum of all small submodules of $M$. My question: "if $x\in rad(M)$ is it true in general that $Rx$ is a small submodule of $M$? ...
1
vote
2answers
50 views

Irreducible polynomials of the form $x^n - q$

Is there any easy way to see the following? Let $n\in\mathbb{N}$. Let $a,b\in\mathbb{R}$ s.t. $a < b$. Then, there exists $q\in\mathbb{Q}$ s.t. $a<q<b$ and $p(x) = x^n - q$ is irreducible ...
1
vote
0answers
21 views

Find a non zero element $z\in \mathbb Z_{100}$ such that $yz = 0_{R}$ and $zy = 0_{R}$ where $y =\overline{ 14}$

Find a non zero element $z\in \mathbb Z_{100}$ such that $yz = 0_{R}$ and $zy = 0_{R}$ where $y =\overline{ 14}$ For this I have found such an element to be $\overline{50}$ since ...
1
vote
0answers
21 views

xor of three relations using Relation Algebra operations

Suppose I have three relations R1, R2, and R3. How can I specify xor of these three relations using relation algebra operations. How this scales up (for example, for four relations)? Thanks I add ...
0
votes
1answer
29 views

Why is that the radical ideal?

In my lecture notes we have the following: Definition: $f, g \in \mathbb{C}[x, y]$ $f \sim g \Leftrightarrow \exists c \in \mathbb{C}, c \neq 0$ such that $g=cf$ Example: If $f \sim g ...
1
vote
1answer
41 views

Example of a linear algebraic group which is not a Lie group

I am trying to reconcile the notions of algebraic groups, linear algebraic groups, Lie groups, and Lie algebras, along with their notions of root systems, maximal tori, etc. To begin, I am trying to ...
2
votes
1answer
83 views

What is an intuitive way to think of Cauchy's theorem?

I am looking at a problem which involves an understanding of why a finite group $G$ has an element with order $p$ if $p$ is a prime factor of $|G|$. I have looked at several resources and proofs ...
2
votes
1answer
36 views

Tensor Algebra: Symmetrization & Antisymmetrization

Problem Given the tensor algebra: $$TV:=\sum_{k=0}^\infty{\bigotimes}^k V$$ Regard the symmetrization and antisymmetrization: ...
0
votes
1answer
31 views

Find the order of $GL_2(Z_{p^{n}})$ for each prime ${p}$ and positive integer ${n}$.

Let $GL_2(Z_m)$ denote the multiplicative group of invertible $2 * 2$ matrices over the ring of integers modulo m. Find the order of $GL_2(Z_{p^{n}})$ for each prime ${p}$ and positive integer ${n}$.
0
votes
0answers
12 views

Evaluating minimal polynomial over a field $F$ as a characteristic polynomial for a $F$-linear map.

I'm considering $K/F$ to be an extension of degree $n$. I've shown that for any $a\in K$, the map $\mu_a : K → K$ defined by $\mu_a(x) = ax$ for all $x\in K$, is a linear transformation of the ...
0
votes
0answers
28 views

Proving that the irreducible polynomial of $\sqrt 2 + \sqrt 3$ over $\mathbb Q$ is irreducible modulo $p$ for every prime $p$. [duplicate]

I've computed the irreducible polynomial of $\sqrt 2 + \sqrt 3$ over $\mathbb Q$ to be $x^4+10x^2+1$. I want to show that this polynomial is irreducible module $p$ for every prime $p$. How do I do ...