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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2answers
55 views

Finding the kernel of maps between (polynomial) rings

If I have a map between rings like $f\colon k[x_1,x_2]\to k[t],x_1\mapsto t^2-1,x_2\mapsto t^3-t$, how can I prove that the kernel is $\mathfrak{a}=(x_2^2-x_1^2(x_1+1))$? I see that ...
2
votes
2answers
24 views

Let $H$ be normal subgroup of $G$. If $G/H$ is cyclic group generated by $aH$, prove that $G=KH$ where $K=\langle a\rangle$.

Let $H$ be normal subgroup of $G$. If $G/H$ is cyclic group generated by $aH$, prove that $G=KH$ where $K=\langle a\rangle $. I would like someone to check my solution. First of i will prove that $G$ ...
-2
votes
1answer
41 views

Algebra, questions on max ideals in rings, and prime ideals [closed]

This is a two part question: If $M \triangleleft_{\max} (R,+,\cdot)$ ($M$ is a max in a commutative, associative ring $R$ with unity.) Then $0_+ \in M$? Why? $R$ commutative, associative ring with ...
4
votes
1answer
40 views

$\mathbb{F}_p[T, 1/T]$ is discrete in $\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T))$, adeles.

Let $p$ be a prime number. How do I show that $\mathbb{F}_p[T, 1/T]$ is discrete in $\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T))$?
3
votes
1answer
45 views

Proof of Cohn's Irreducibility Criterion

I was looking for an elementary (or involving introductory level abstract algebra/analysis) proof of Cohn's Irreduciblity Criterion: If $$ a_0, a_1, \dots, a_n \in \Bbb{Z} $$ and $$ 0 \le ...
2
votes
1answer
44 views

What would be interesting maps to use on that Eudoxus reals?

I'm trying to understand Eudoxus Reals. From wikipedia: Let an almost homomorphism be a map $f:\mathbb{Z}\to\mathbb{Z}$ such that the set $\{f(n+m)-f(m)-f(n): n,m\in\mathbb{Z}\}$ is finite. We say ...
4
votes
1answer
58 views

Is there a name for this simple structure?

Is there a name for $(X,S)$ where $X$ is a set and $S\subseteq X$ and a morphism $(X,S)\overset{\alpha}{\longrightarrow}(X^\prime,S^\prime)$ is a function $\alpha:X\rightarrow X^\prime$ such that ...
2
votes
1answer
50 views

If $\Delta$ is cocommutative in $k(G)$, why does that imply $G$ is abelian?

Suppose $G$ is a finite group and $k(G)$ is it's group function Hopf algebra. I read that for $k(G)$ is quasi-triangulated requires that $G$ be abelian. If $R$ is the distinguished element of ...
1
vote
0answers
81 views

How coefficients from finite field can form ring of polynomials?

Let us consider a graph $G(V,E),$ where $V$ is the set of nodes and $E$ is the set of edges. $\mathbf{x}=[X_1,\ldots,X_r]$ are symbols multicast by source to $|T|$ sink nodes. Symbols are from ...
4
votes
1answer
71 views

If $m$ divides $n$, find a free resolution of $\mathbb{Z}/m$ as a $\mathbb{Z}/n$-module.

If $m$ divides $n$, find a free resolution of $\mathbb{Z}/m$ as a $\mathbb{Z}/n$-module. I have tried this one and got $0 \leftarrow \mathbb{Z}/m \leftarrow \mathbb{Z}/n \leftarrow \mathbb{Z}/n$. ...
1
vote
0answers
47 views

Prove that $n \in \mathbb{Z}^\star \Leftrightarrow n \mid 1$ and $n-1 \mid 1$ or $n+1 \mid 1$, and $(x-1)/(t-1) \equiv n \pmod {t-1}$

I want to prove the following lemma: Let $F[t,t^{-1}]$ be the ring of the polynomials in $t$ and $t^{-1}$ with coefficients in the field $F$ and assume that the characteristic of $F$ is zero. ...
0
votes
1answer
26 views

In a modular arithmetic equation, how to find 'a' given a range?

Say I have an equation in the form $$a\bmod b = c$$ I know $b$ and $c$ I'm given a range $(d,e)$ (where $d$ and $e$ are integers). How can I find all values of $a$ that satisfy the inequality $d ...
0
votes
2answers
65 views

Prove that an ideal $ \mathfrak{m} $ of a commutative ring $ R $ is maximal iff $ R/\mathfrak{m} $ is simple.

Could someone give me a hint on whether I’m on the right track or not? For sufficiency, I tried the following: Suppose that $ \mathfrak{m} $ is a maximal ideal. With the quotient map, we get $ ...
2
votes
1answer
27 views

Basis for the field extension $\mathbb{Q}(\zeta_{12})$

Consider the cyclotomic field $\mathbb{Q}(\zeta_{12})$ where $\zeta_{12}$ represents the $12$-th primitive root of unity. Since the minimal polynomial of $\zeta_{12}$ is given by $\Phi_{12}(x)$ which ...
-2
votes
0answers
44 views

Chain of three prime ideals of $\mathbb F_p[x,y]$ [closed]

Let $A_1\neq\left \{ 0 \right \} $, $A_2$, and $A_3$ be prime ideals of $\mathbb{F}_p[x,y]$ such that $$A_1\subset A_2\subseteq A_3\subset \mathbb F_p[x,y]$$ Then $A_2 = A_3$.
0
votes
1answer
41 views

What's a diagonal sum of two matrices?

Let $A$ be an $n\times n$ matrix over a field $K$. Show that there exists an invertible matrix $P$ such that $P^{-1}AP$ is a diagonal sum of an invertible matrix and a nilpotent matrix. (Hint: use ...
2
votes
2answers
104 views

How to prove the group $G$ is abelian?

Question: Assume $G$ is a group of order $pq$, where $p$ and $q$ are primes (not necessarily distinct) with $p\leqslant q$. If $p$ does not divide $q-1$, then $G$ is Abelian. I know that if the ...
1
vote
1answer
69 views

Prove that in the ring $F[t,t^{-1}]$ we have $x=t^n \Leftrightarrow x \mid 1$ and $t-1 \mid x-1$

I want to prove the following lemma: For any $x$ in the ring $F[t,t^{-1}]$ ($F[t,t^{-1}]$: the polynomials in $t$ and $t^{-1}$ with coefficients in the field $F$), $x$ is a power of $t$ if and ...
1
vote
0answers
34 views

$x^p -x-1$ irreducible over $\mathbb{F}_{p}$ [duplicate]

Show that $x^p - x -1$ is irreducible over $\mathbb{F}_{p}$. I've seen this polynomial (or some variation x^p -x -a) on several of our qualifying exams and in every case they ask you to show it is ...
4
votes
2answers
65 views

“If $x$ is a non-unit, then $1-ux$ is a unit”

I don't understand these two lines from my book. We are given that $R$ is a local ring. If every $2$-generator submodule is cyclic and $Ra$, $Rb$ are given, then $Ra+Rb=Rc$, hence ...
0
votes
1answer
33 views

What's a $2$-generator submodule?

Context: Let $R$ be a local ring and $M$ an $R$-module. Show that the set of all submodules of $M$ is totally ordered by inclusion iff every finitely generated submodule of $M$ is cyclic or, ...
2
votes
1answer
18 views

Assumption on characteristics in an exercise about roots of unity

I'm solving the following exercise: "Let $K$ be a field, $char(K) \nmid 2n$ for $n \geq 1$ an odd integer. If $K$ contains a primitive $n$-th root of unity, then it also contains a primitive $2n$-th ...
5
votes
2answers
62 views

Show that $f$ is a homomorphism.

There is a group $G$ of order $p^3$, where $p>2$. Show that $f:G\rightarrow Z(G) $ with $f(x)=x^p$ is a homomorphism. My attempt: Case a): Suppose $|Z(G)|=p^3$. Then $G=Z(G)$, so $G$ is abelian, ...
-1
votes
0answers
26 views

Why is $Z[X]/<5,X^3 + X + 1>$ a field? [duplicate]

Prob. Show that $Z[X]/<5,X^3+X+1>$ is a field. Since $Z[X]$ is a commutative ring, it is sufficient to prove that $<5,X^3+X+1>$ is a maximal ideal. Since $5$ and $X^3+ X +1$ are ...
0
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0answers
38 views

Exponential map only for matrix Lie algebras?

Recently, I stumbled over some proofs in Lie algebra theory and noticed that they often use the notion of an exponential map $e^{t \zeta}$ for $\zeta \in \mathfrak{g}$ such that $e^{t \zeta} \in G$ ...
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votes
0answers
32 views

How to prove the theorem on group algebra of permutation group [closed]

How to prove the following theorem: If $t$ is a vector in group algebra of permutation group, then $\cal{y}t\cal{y}=\lambda_t \cal{y}$, where $\cal{}y$ is the Young operator of permutation group and ...
4
votes
1answer
37 views

group-like elements in the Hopf algebra of the homology of $H$-spaces

Let $X$ be an $H$-space with product $\mu$. Then $H_*(X)$ is a Hopf algebra with product $\mu_*$. Let $\psi$ be the coproduct of the Hopf algebra $H_*(X)$. Define a subset $S$ of $H_*(X)$ as $$ ...
3
votes
1answer
102 views

Subring of $M_7(\mathbb{Z}_2)$ isomorphic to $\mathbb{F}_{128}$?

Let $A \subset M_7(\mathbb{Z}_2)$ be a subring such that no proper nonzero subgroup $V \subset \mathbb{Z}_2^7$ is invariant under all matrices in $A$. I suspect that $A \cong \mathbb{F}_{128}$, but ...
4
votes
2answers
84 views

Algebraic proof that a set generated by irrational rotations is dense in $S^1$.

This is exercise 1.9 in Lie Groups, Lie Algebras and Representations - Hall. Suppose $a$ is an irrational real number. Show that the set $E_a$ of the numbers of the form $e^{2\pi i n a}$, $n \in ...
2
votes
3answers
65 views

Defining a coproduct in $\mathsf{Grp}$ using group presentations

I've encountered this exercise in Aluffi's Algebra: Chapter 0. It might be helpful to say that the book doesn't introduce functors at this stage,, and that the book defines a presentation of a group ...
1
vote
1answer
27 views

A question about fields and separability in Serre's “Local Fields”

On page 14 of the English edition of Serre's "Local Fields", that is chapter 1, section 4, I am confused by the following; there is talk of fields $B/\mathfrak P$ and $A/\mathfrak p$ for prime ideals ...
1
vote
1answer
25 views

Lie algebra of affine linear maps

Let $G$ be the Lie group of affine transformations, $$\{x \mapsto Ax+b,A \in GL(n), b \in \mathbb{R}^n\}.$$ We can represent these maps as matrices $$\begin{pmatrix} A & b \\ 0 & 1 ...
1
vote
0answers
39 views

Number of conjugacy classes of nonabelian group of order $pq$.

The problem asks to show that a nonabelian group of order $pq$ has $p+\frac{q-1}{p}$ conjugacy classes. I have shown a. $p$ divides $q-1$, b. $|Z(G)| = 1$, Now I'm using the class equation to ...
2
votes
0answers
33 views

Tangent space of coadjoint orbit

Let $\xi \in T_xOx$ be a tangent vector at $x \in O_x :=\{\mathrm{ Ad} _{u}^*(x); u \in G\}$ for $x \in g^*.$ ($g$ is the Lie-Algebra) Then I read that this $\xi$ can be represented as the velocity ...
3
votes
3answers
45 views

Why is $\phi : \operatorname{Hom}(\mathbb{Z},G) \to G$ given by $ f \mapsto f(1)$ surjective?

I was working on showing $\operatorname{Hom}(\mathbb{Z},G) \cong G$ for $G$ abelian. The proposed map given by evaluating a given $f \in \operatorname{Hom}(\mathbb{Z},G)$ at $1$ is easily seem to be a ...
2
votes
0answers
27 views

$A$ simple algebra $\implies$ $\mathcal M(A)$ multiplication algebra simple?

I would like to know whether the following holds: If $A$ is a simple, non-associative algebra over a field $K$, then the multiplication algebra $\mathcal M(A)$ (see below for definition) is simple ...
-1
votes
0answers
17 views

Lie algebra affine transformations [duplicate]

Let $G$ be the Lie group of affine transformations $$\{x \mapsto Ax+b,A \in GL(n), b \in \mathbb{R}^n\}.$$ Then we can represent these maps as matrices $\begin{pmatrix} A & b \\ 0 & 1 ...
4
votes
2answers
45 views

In a group $G$, prove the following result

Let $G$ be a group in which $a^5=e$ and $aba^{-1}=b^m$ for some positive integer $m$, and some $a,b\in G$. Then prove that $b^{m^5-1}=e$. Progress $$aba^{-1}=b^m\Rightarrow ab^ma^{-1}=b^{m^2}$$ ...
0
votes
0answers
15 views

Adjoint and coadjoint orbits

I just read that for the Lie algebras $\mathfrak{gl}(N),\mathfrak{sl}(N),\mathfrak{so}(N),\mathfrak{sp}(2N)$ the adjoint and coadjoint orbits coincide. Now, the adjoint orbits are $O_{\xi} = ...
0
votes
1answer
46 views

Killing form - strange definition

I was just reading about Killing forms. In my opinion, the definition of these forms is quite strange. I mean why would one define $B(X,Y) = \mathrm{tr} (\mathrm{ad} (X) \circ \mathrm{ad} (Y))$? I ...
2
votes
1answer
24 views

How do we define discriminant over a commutative ring?

Let $f$ be a nonconstant polynomial over a field $F$. Since there exists a splitting field of $f$ over $F$, $f$ can be decomposed as $f=c\prod_{i=1}^n (X-\alpha_i)$ Hence, it is possible to define ...
2
votes
1answer
24 views

Class equation question from Artin's book.

Let G be a group of order n that operates nontrivially on a set of order r. Prove that if n > r!, then G has a proper normal subgroup. Also I am not very clear of the term "operates nontrivially", ...
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votes
0answers
30 views

The learning problem of abstract algebra [closed]

what are the learning problems of abstract algebra and their measures?
2
votes
2answers
66 views

Conditions for an orthogonal matrix equation

Let $B_1$ and $B_2$ be given $n \times n$ real non-singular matrices and consider the system of equations $$\begin{bmatrix}A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}\begin{bmatrix}B_1 ...
4
votes
1answer
115 views

Question concerning the group over GL$(n,\mathbb{Z})$

Is every vector $[a_1,a_2,\dots, a_n]$ with $\gcd(a_1,a_2,\dots,a_n)=1$ a column in some matrix $A\in GL(n,\mathbb{Z})$? I don't think this is a duplicate: Let me rephrase this questions using ...
2
votes
1answer
98 views

Proof: $\mathbb{Z}[\zeta_6]$ is a PID.

I am reading through A First Course in Modular Forms. In Proposition 2.2.3 they claim that $\mathbb{Z}[\zeta_6]$ is known to be a principal ideal domain. Does anyone have a reference for the proof of ...
1
vote
0answers
26 views

Conjecture concerning involutions in a unitary braided fusion category/Grothendieck ring

Despite the categorical setup, a solution to this question may require no categorical tools (see Conjecture 2). Let $\mathcal C$ be a unitary braided fusion category, $I$ be its set of isomorphism ...
1
vote
3answers
74 views

What does $\overline{r}m:=rm$ mean?

On this Wikipedia article, it says that you can define an $R$-module $M$ as an $R/Ann_R(M)$-module using the action $\overline{r}m:=rm.$ What does that action actually mean? What is $\overline{r}$?
3
votes
1answer
28 views

Adjoint representation is Lie algebra homomorphism

Let $T_g:=L_g R_{g^{-1}}: G \rightarrow G$ be the standard automorphism of a Lie algebra, then $Ad_g:=DT_g(e): \mathfrak{g} \rightarrow \mathfrak{g} $is called the adjoint representation. Now, I want ...