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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Irreducible component of a scheme over a dvr

Let $\mathcal M$ be a (reduced) quasi-projective scheme over a dvr (of mixed caracteristics), $R$. Suppose that the generic fiber $\mathcal M_{\eta_R}$ is (nonempty) smooth and irreducible of ...
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1answer
15 views

How do I construct a nontrivial linear representation of the group $G$=S' in $R^3$

$S'$ is defined as the unit circle. The product is defined as the sum of angles. How do I construct a linear representation in general? I don't know how to begin with this problem. Some ...
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38 views

Number of all possible groups of given order [duplicate]

Suppose $n=18$, then all possible groups of order $18$ is $5$. Among them $2$ are abelian and $3$ are non-abelian. Let $n$ be a natural number. How can I determine the number of all possible ...
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14 views

Difference between torsion and zero divisor

I'm not understanding what the difference is between a zero divisor and a torsion element of a module. My best guess is that the torsion elements are "vectors" and zero divisors are scalars. This ...
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1answer
32 views

If $gcd(a,b)=1$, then there exists integers x and y such that $xa + yb = 1$

Did not find this from this website... If $$ gcd(a,b)=1,$$ then there exists integers x and y such that $$xa+yb=1.$$ Now, the tip is to use particular corollary, that states: The class $[m]_{n}$ ...
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1answer
12 views

Show that there exists $\mathbb{Z}[i]$ modules $M$ and $N$ such that both have $13$ elements but are not isomorphic

$\textbf{Question:}$ Show the existence of two $\mathbb{Z}[i]$ modules $M$, $N$ such that both $M$ and $N$ have $13$ elements bot $M \not \cong N$ as $\mathbb{Z}[i]$ modules. $\textbf{My Attempt:}$ ...
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2answers
40 views

Is $(x) \otimes_{\frac{k[x]}{(x^2)}} (x)$ zero?

I am trying to decide if $(x) \otimes_{\frac{k[x]}{(x^2)}} (x)$ is zero. So I considered $x \otimes x$ which I rewrote as $1 \otimes x^2 = 1 \otimes 0 = 0$. But then I realized that $1$ does not live ...
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54 views

Automorphism of $D_6$

I need find an explicit way to express the group $Aut(D_6)$, and I have not idea how write this group, maybe this is an semidirec product of some groups but I don´t see this. thanks.
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11 views

Why does $\operatorname{Ad}_h((S\otimes 1)(Q))=\epsilon(h)(S\otimes 1)(Q)$ in a quasi-triangular Hopf algebra?

I'm reading a proof that in a quasi-triangular Hopf algebra $H$, $(S\otimes 1)Q$ is $\operatorname{Ad}$-invariant. Here $Q=\tau(R)R$, where $R$ is the invertible element in $H\otimes H$ satisfying all ...
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1answer
35 views

Tangent space of quotient space

Let $\pi : M \rightarrow M/G$ be the canonical projection, where $M$ is a manifold and $M/G$ is a quotient manifold. Now, what can we say about $d \pi (p) : T_pM \rightarrow T_p(M/G)$? From my ...
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51 views

Fields of Research in Algebra [on hold]

I'm a last-year student in mathematics and I'm looking for a master degree in algebra. So I'm trying to understand what are the most interesting fields of research in algebra all around the world. ...
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1answer
24 views

Lie group and stabilizer quotient

Let $G$ be a Lie group and $$G_x:=\{g \in G; Ad^*_g(x)=x\}$$ the stabilizer, where $Ad_g^*$ is the adjoint of the adjoint representation. Now, I was wondering why $G/G_x$ has a manifold structure. ...
3
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1answer
32 views

Global Dimension of a Ring and its Localizations

Why is the following true? The global dimension of a noetherian ring $A$ is the supremum of the global dimension at its localizations at its maximal ideals: ...
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51 views

index 2 subgroups of the infinite product of Z/2Z

Is it possible to describe all the index 2 subgroups of the group $G = \prod_{i\in \mathbb{N}}\; \mathbb{Z}/2\mathbb{Z}$? For example, one can take the kernel of the $i$-th projection map ...
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27 views

Open subscheme in special fiber

Let $X$ be a projective scheme over $R$ a discrete valuation ring with generic fiber irreducible. Can an open subscheme of $X$ be contained in the special fiber of $X$ ? Or is it true that every open ...
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2answers
37 views

Permutations: Interpreting Cycle Notation

I have a problem in interpreting permutation. I think the definition and my interpretation of it don't match each other. Let $\sigma=(1\ 2\ 4\ 3)$, and $\tau=(1\ 3\ 2\ 4)$ in one-line notation. I ...
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1answer
51 views

proving that the symmetric group $S_x$ is not finitely generated where $x$ is infinite [on hold]

it seems pretty trivial, but I have trouble of showing it. I also wonder of good approach of proving that a group is not finitely generated.
2
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1answer
14 views

Commutative Monoid - matrix set

Let $M$={$\begin{bmatrix} a & b & c \\ c & a & b \\ b & c & a \end{bmatrix}|a,b,c\in \mathbb{R}, a+b+c=0$}. The matrices in $M$ are a special kind of Toeplitz matrices ...
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28 views

how is the formula for solving cubic equation derived directly from sum of roots, product of roots and sum of product of roots taken 2 at a time?

In $ax + b = 0$, let the root be $p$. We know $ p = -b/a$ In $ax^2 + bx + c = 0$, let the roots be $p$ and $q$. We know, $$\frac{-b}{a} = p+q $$ $$\frac{c}{a} = pq $$ Now, ...
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36 views

Proving that $t^{p^r}-a$ is irreducible when $a\in k$ is not a $p$th power

Let $p$ be an odd prime, $F$ a field of characteristic $0$ and $a\in F$ with $a\neq 0$. Assume $a$ is not a $p$th power in $F$. Prove that for every positive integer $r$, $t^{p^r}-a$ is irreducible ...
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1answer
38 views

Is it true that every prime ideal of height one is principal? [on hold]

Is it true that every prime ideal of height one is principal ? Please help
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2answers
43 views

$R$ a local ring, also a PID. $I,J$ ideals from $R$. Show that $I \subseteq J$ or $J \subseteq I$

$R$ a local ring, also a PID. $I,J$ ideals from $R$. Show that $I \subseteq J$ or $J \subseteq I$ My brief attempt to try use Bezout theorem at a PID. but unsuccess.. Thanks any help.
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73 views

Is there an infinite topological meadow with non-trivial topology?

For reference meadows are a generalization of fields that were designed to be compatible with the requirements of universal algebra. Specifically a meadow is a commutative ring equiped with an ...
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Comments about “Topics in Algebra” by I.N. Herstein and “Abstract Algebra” by Dummit/Foote? [on hold]

Today, I got two gifts from my research mentor: "Topics in Algebra" by I.N. Herstein and "Abstract Algebra" by Dummit/Foote. I am very happy and grateful for his gifts, but I already have been ...
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1answer
32 views

The difference between Quotient Set and other definition

it a new course and material that I learn, we defined 2 pretty similiar definitions and I didnt understand what is the difference between the definitions. Definition 1: A subset $T\subseteq X$ is ...
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2answers
51 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 ...
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2answers
22 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 $G$ ...
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1answer
37 views

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

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 ...
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1answer
26 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 $\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T))$?
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1answer
43 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 ...
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1answer
42 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 ...
3
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1answer
41 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
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1answer
36 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 ...
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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 ...
3
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1answer
62 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 $0 \leftarrow \mathbb{Z}/m \leftarrow \mathbb{Z}/n \leftarrow \mathbb{Z}/n$. ...
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46 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. ...
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1answer
24 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 ...
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2answers
64 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 $ ...
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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 ...
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Chain of three prime ideals of $\mathbb F_p[x,y]$ [on hold]

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 $A_2 = A_3$.
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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 ...
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2answers
103 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 ...
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1answer
63 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 ...
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0answers
33 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
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2answers
62 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 ...
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1answer
32 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, ...
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1answer
17 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 ...
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2answers
61 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, ...
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25 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 ...