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

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4
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1answer
60 views

Which group is isomorphic to?

If I have an abelian group $G$ of order $p^n$, how can I decide if it's isomorphic to $\Bbb{Z}_p \times \Bbb{Z}_p \times\ldots \times \Bbb{Z}_p$ ($n$ times) or to $\Bbb{Z}_{p^2} \times \Bbb{Z}_p \...
-2
votes
1answer
27 views

Field extension $\mathbb F_p\subset E$ [closed]

Suppose there exists a field extension $\mathbb F_p\subset E$. Question: Is it possible that the degree is $[E:\mathbb F_p]=2$. And how many elemnts are in E then? How can I proof such a question?...
1
vote
1answer
46 views

Exercise on the ring $\mathbb Z \times \mathbb Z$ and its quotient with an ideal

Let $A = \mathbb Z \times \mathbb Z$ a ring, where operations are defined elementwise. a) Prove that the ideal $I$ generated by $x = (4,6)$ is not maximal. b) Find in $A$ (if it exists) an ...
0
votes
1answer
50 views

Finite type + integral = finite

Let $A \subseteq B$ be rings (comm. with unity). I am struggling to see why the following equivalence holds for $B$ interpreted as a $A$-Algebra: $A \rightarrow B$ is of finite type and $A\...
4
votes
1answer
39 views

Does the commutator group of $S_n$ equal $A_n$ in general?

And how would one deduce this? $[S_n, S_n]$ consists of even permutations so it's obvious that $[S_n, S_n] \leq A_n$, but is $[S_n, S_n] = A_n$ true as well? If so, how to deduce this? If not, how ...
7
votes
0answers
45 views

Maximal ideals in the ring of measurable functions

The $R$ ring of continuous functions from $[0,1]$ to $\mathbb{R}$ has a property that its maximal look like a subset of $R$ consisting of those functions which vanish at a common single point in $[0,...
1
vote
1answer
41 views

If $u, v$ have different minimal polynomials, then $F(u)$ is not isomorphic to $F(v)$?

Is the following true? Let $F$ be a field. Suppose $u,v$ have different minimal polynomials $p_u,p_v\in F[X]$, then $F(u)$ is not isomorphic to $F(v)$ as fields. I am asking this because I ...
-1
votes
0answers
31 views

Principal Ideal in a polynomial ring [duplicate]

Let $K[x]$ be a polynomial ring. If I am given two polynomials $P_1$ and $P_2$, and if I find the generator of the ideal of those two polynomials, How can I tell whether or not that ideal is principal?...
2
votes
2answers
38 views

Given a ring $R$ and a ring extension $R'$, if $r=r's$ where $r'\in R'\setminus R$, does that mean that $s\not\mid r$ in $R$?

Given a ring $R$ and a ring extension $R'$, if $r=r's$ where $r'\in R'\setminus R$ and $r,s\in R\setminus\{0\}$, does that mean that $s\not\mid r$ in $R$? I was thinking for example in $\Bbb{Z}$, ...
0
votes
2answers
88 views

Is the ring $m\mathbb{Z}$ isomorphic to the ring $n\mathbb{Z}$?

I came over a question in ring theory which I am not being able to proceed upon: When is the ring $m\mathbb{Z}$ isomorphic to the ring $n\mathbb{Z}$, where $m, n \in \mathbb{N}$? I know that to ...
0
votes
1answer
71 views

Monic irreducible polynomials over infinite field

If $F$ is a countable field, then proving that $F$ has algebraic closure is quite simple: there can be at most countable number of monic irreducible polynomials over $F$, let they form the set $\...
2
votes
3answers
51 views

Showing a finite abelian group is cyclic assuming something about all homomorphic images of it

Let $G$ be a finite abelian group such that $|G|\ne p^n$ for any prime $p$. If every homomorphic image $\varphi (G)$ with $|\varphi (G)| < |G|$ is cyclic, then show $G$ is cyclic. This is an old ...
1
vote
1answer
103 views

Prove that $\Bbb{R}[\cos(\theta),\sin(\theta)]\cong\Bbb{R}[x,y]/(1-x^2-y^2)$ [duplicate]

More precisely, given the ring homomorphism $\phi:\Bbb{R}[x,y]\to\Bbb{R}^\Bbb{R}$, with $\phi(f(x,y)):\Bbb{R}\to\Bbb{R},\,\,\phi(f(x,y))(\theta)=f(\cos(\theta),\sin(\theta))$, where $\Bbb{R}[x,y]$ is ...
5
votes
3answers
52 views

The group algebra $KG$

If $G$ is a cyclic group of order $m$. Then $KG\cong K[t]/(t^m-1)$. Where $K$ is a field. I define \begin{align*} \varphi:K[t]&\longrightarrow KG\\ \sum_ia_it^i&\longmapsto\sum_ia_ig^i \end{...
1
vote
1answer
31 views

nilpotent ideal of incidence algebra

Let $(I;\preceq)$ be a finite poset, where $I=\{a_1,\ldots,a_n\}$ and $\preceq$ is a partial order on $I$. The subset $$KI=\{\lambda=[\lambda_{ij}]\in\mathbb{M}_n(K)\mid\lambda_{st}=0\text{ if } a_s\...
2
votes
0answers
60 views

What is difference between the “usual multiplication” and multiplication?

I have been reading books in the algebra, and I noticed that some books use terms "usual multiplication" and "usual addition". Do they carry different meaning that multiplication and addition? If "...
4
votes
0answers
46 views

On kernels of commuting operators in infinite dimensions

Let $X$ be an infinite dimensional vector space, and let $\operatorname{S},\operatorname{T}\in\mbox{End}(X)$ be two operators such that: $\operatorname{T}\operatorname{S}=\operatorname{S}\...
4
votes
4answers
170 views

Element of infinite order for a given group presentation

Let $G=\langle a,b,c,d \mid abcda^{-1}b^{-1}c^{-1}d^{-1}\rangle$ be our presentation. The claim is that the commutator $[a,b]$ has inifinite order in $G$. I think this might be related to small ...
0
votes
1answer
23 views

Is it true that the only regular elements in $Z_m$ are invertible ones?

I have this doubt. In a unitary and commutative ring $$Z_m = \{[0]_m, [1]_m,\ ...\ ,\ [m - 1]_m\}$$ There are only two "kind" of elements: invertible and zero divisors. Is it true to say that the ...
1
vote
1answer
20 views

Empty singular submodule

I search for a module $M$ with its singular submodule $Z(M)$ the empty set, i.e. for every element $m$ of $M$ the annihilator of $m$ in $R$ is not essential, say, as right $R$-module; or proving that ...
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votes
1answer
65 views

Help finding an article [closed]

Hello Recently I have been studying algebra and am in search of the following paper : Kac, V. G. Classification of simple $Z$-graded Lie superalgebras and simple Jordan superalgebras. Comm. Algebra 5 ...
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0answers
26 views

Function check-up exercise

I want to make sure I did everything correctly, so here's the exercise: Given $P$ the set of positive prime numbers and be $S = \mathbb N^* - \{1\}$. $\forall n \in S,\ \pi(n)$ is the set of the ...
1
vote
1answer
64 views

Extension of Scalars is well-defined

The reason I'm asking this, is because as an exercise, I'm asked to prove the following: Let $A$, $B$ be rings, $f:A\to B$ a ring homomorphism inducing $A$-module structure on $B$, and $M$ a flat $A$-...
1
vote
1answer
32 views

If $N$ is a normal subgroup of $G$ then not necessarily $\varphi(N)$ is a normal subgroup of $G'$

Let $\varphi: G \longrightarrow G'$ be a homomorphism group. If $N$ is a normal subgroup of $G$ then not necessarily $\varphi(N)$ is a normal subgroup of $G'$. Of course the Ker $\varphi$ and Im $\...
4
votes
2answers
64 views

Motivation for the study of algebraic structures

I am currently studying group theory and I realized that most concepts we study are just definitions on which we build theory. I do understand that some theorems are beautiful and don't need any ...
2
votes
1answer
69 views

If $G=\left<(12),(34),(45)\right>\subset S_5$, then $G\cong C_2\times S_3$

Let $G=\left<(12),(34),(45)\right>\subset S_5$. Show that $G\cong C_2\times S_3$. So my first idea was to set $a=(12)$, $b=(34)$ and $c=(45)$ and remark that $$G=\left<a,b,c\mid ab=ba,ac=ca, ...
3
votes
0answers
33 views

What are the units of $U(\mathfrak{sl}_2)$?

Let $U(\mathfrak{sl}_2)$ be the Universal Enveloping Algebra of $\mathfrak{sl_2}$ over a field $K$, i.e. the (non-commutative) algebra generated by three generators $E,F,H$ subject to the commutator ...
0
votes
1answer
31 views

Prove that any finitely generated submodule of $R^+$ (the field of quotients) is free of rank $1$

I am working on the following problem: Let $R$ be a principal ideal domain and $R^+$ the field of quotients. Then $R^+$ is an $R$-module. Prove that any finitely generated submodule of $R^+$ is a ...
1
vote
1answer
47 views

$\mathbb{Z}_p$ is an Integral Domain

Assume that we define the ring of the $p$-adic integers as the projective limit $$\mathbb{Z}_p =\varprojlim \frac{\mathbb{Z}}{p^n\mathbb{Z}}$$ Then $\mathbb{Q}_p$, the field of the $p$-adic numbers ...
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votes
0answers
22 views

Mandelbrot set and times tables

I recently saw a mathologer video on YouTube titled Times tables, Mandelbrot set and the heart of mathematics. It was about generating patterns using tables of numbers. I don't have any idea about it. ...
0
votes
0answers
28 views

Given a homomorphism defined on a generating set of a group how to define it for a general element?

Let $\mathbb Z^n$ and $\mathbb Z^d$ be free $\mathbb Z$-modules with $d>n$. Suppose $v_1,\dots,v_d$ are primitive vectors in $\mathbb Z^n$. Let $e_1^*,\dots, e^*_n$ be the dual basis (that is, a ...
4
votes
1answer
82 views

Number of subgroups equal to order of group

Here is a fun question. Consider the dihedral group $\mathcal{D}_4=\left\langle a,b\mid a^4=b^2=1, bab=a^{-1}\right\rangle$ of order $8$. This group has exactly $8$ genuine subgroups (but not all ...
0
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0answers
35 views

Are there groups $G$ isomorphic to $\mathrm{Aut}\left( \mathrm{Aut} (G)\right)$, such that $G\not\cong \mathrm{Aut}(G)$?

Trivial examples for the first condition are easy to find: $G={1},C_2$. Are there any (finite\non-finite) groups that satisfy the conditions in the title?
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0answers
48 views

Finding square root of polynomial in extension field (Number field sieve)

I was reading this paper, on page number 29, 2nd paragraph it is written that "take the coefficients of $ \gamma $ modulo q and applying an algorithm for taking square roots in the finite field Z[ $ \...
0
votes
0answers
71 views

Homology Groups of Tangent 2-Spheres

I have been trying to compute the Homology Groups $ H_n $ of two tangent 2-Spheres (we will call this space, X). By previous results, I am able to easily determine that $ H_0(X) $ is isomorphic to the ...
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1answer
30 views

A question on part of the proof of the theorem that if $R$ is a UFD then $R[x]$ is a UFD as well.

I have a question regarding a proof in Peter Falb's Methods of Algebraic Geometry in Control Theory, volume I, for the claim in the title. On pages 16-17 he proves the property (ii) of UFDs that the ...
2
votes
0answers
52 views

Problem sets on Abstract Algebra

Many times we ask about what books should we read to learn or know more about a math topic (Abstract Algebra, in this case). However, I would like to get a list of the exercises what should we solve ...
1
vote
1answer
22 views

Name for submodule killed by a right ideal

Let $\mathfrak a$ be a right ideal in a ring $R$. The set $N=\{m\in M: \mathfrak am=\mathfrak 0\}$ is a submodule of the left $R-$module $M$: If $m,n\in N$, $a\in \mathfrak a$, then $a(m-n) = am-an =...
3
votes
3answers
79 views

Can the dihedral groups be seen as subgroups of each other?

I am solving one difficult problem and now I need information on this: If $m\mid n$, is then possible to "embed" $D_m$ to $D_n$, or otherwise said, does $D_n$ have a subgroup isomorphic to $D_m$? ...
2
votes
2answers
267 views

Algebraic structure on any infinite set

Given any algebraic object $X$, say group, ring, integral domain, etc., and a special subset $I$ of $X$ namely normal subgroup, ideal etc., it is always possible to put a structure on $X/I$ induced ...
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vote
1answer
41 views

Quotient ring with reducible polynomial

Let $S = \mathbb{R}[x]/(x^2+1)^2).$ The first goal is to show that there exist exactly two homomorphisms $ \pi\colon S\to \mathbb{C} $ such that $\pi|_{\mathbb{R}} = \text{id}_{\mathbb{R}}.$ I know ...
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vote
2answers
50 views

Factoring $p(x) = x^n -1$ for any natural number $n$

Can I say that from inspection, $(x-1)$ is a factor which implies that $p(1) = 0$. I then used long division which then gave this: $p(x) = (x-1)(\sum \limits _{i=1} ^{n} x^{n-i})$ How would you have ...
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0answers
45 views

I am confused about this notation for matrix representation.

I am confused about this notation from the image below. How can you represent something like (1,2),1 as a 2D matrix? For S1 = (0,1,2) and S2 = (0,1) I would expect two matrices like: [(1,0),(1,0),...
0
votes
1answer
56 views

$17\mathbb{Z}[\sqrt {10}]$ is prime ideal in $\mathbb{Z}[\sqrt {10}]$

This seems tedious since I would have to show $(a+b\sqrt {10})(c+d\sqrt {10})=17k+17j\sqrt {10}$ implies that one of the factors belongs to $17\mathbb{Z}[\sqrt {10}]$. It's easier to show something ...
1
vote
1answer
70 views

Show that $V=\mathbb KG\oplus\cdots\oplus \mathbb KG$.

Let $V$ a $\mathbb KG$-module such that the character of $V$ is such that $\chi_V(g)=0$ for all $g\in G\setminus \{1\}$. Show that there is an $m$ s.t. $$V=\underbrace{\mathbb KG\oplus\cdots\oplus \...
2
votes
2answers
76 views

Sylow subgroup of a product of subgroups

Let $UV$ be a subgroup of $G$ and $P$ is a Sylow $p$-subgroup of $G$. Suppose that $P \cap U \in \operatorname{Syl}_p(U)$ and $P \cap V \in \operatorname{Syl}_p(V)$. I need to show that $P\cap UV \in \...
2
votes
0answers
44 views

Automorphism of $\mathbb C(x,y)$ and its order in $\mathrm{Aut}(\mathbb C(x,y))$

In the following problem Let $M=\bigg (\begin{matrix} a& b\\ c& d\end{matrix}\bigg )$ be a nonsingular matrix with integer coefficients and $L=\mathbb C(x,y)$. (i) Show that $\phi(x)=...
0
votes
2answers
38 views

Properties of a subgroup of a group $\mathbb Z_p \times \mathbb Z_p$

Let $p \geq 5$ be a prime. Thhen which one of the followings are true. 1) $\mathbb Z_p \times \mathbb Z_p$ has atleast five subgroup of order p. 2) Every subgroup of $\mathbb Z_p \times \mathbb Z_p$ ...
0
votes
0answers
38 views

Character group and lattices

Let $\Lambda$ be the complex n-th dimensional lattice over Eisenstein integers ($\mathbb{Z}[\omega]$)). The map $R: \mathbb{C} \mapsto \mathbb{R}$ is defined as following: $R(z)=R(z_{a}+\omega z_b)=...
0
votes
0answers
29 views

Finding an essential submodule

Let $R$ be a commutative ring with unity, and let $M$ be a unitary faithful $R$-module. Assume the annihilator $A$ of $r\in R$ is essential in $R$ (as an $R$-module). I search for an essential ...