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Questions tagged [abstract-algebra]

For questions about monoids, groups, rings, modules, fields, vector spaces, algebras over fields, various types of lattices, and other such algebraic objects. Associate with related tags like [group-theory], [ring-theory], [modules], etc. as necessary to clarify which topic of abstract algebra is most related to your question and help other users when searching.

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Determining If Set is Ring or Field

Decide whether the following subset of $\mathbb{R}$ is not a ring, or is a ring but not a field, or is a field: S = {${a + b\sqrt[3]{2} + c\sqrt[3]{4} | a,b,c \in \mathbb{Q} } $ } I tried approaching ...
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0 answers
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Question regarding the meaning of a theorem about characters (of finite abelian groups)

I am trying to understand the meaning behind a theorem regarding characters. First, here are the basic definitions and notations. Let $G$ be a finite abelian group. Define $\hat{G}:=\{\chi:G \...
1 vote
1 answer
31 views

Direct sum of free group and quotient of abelian group by subgroup

I'm currently studying abelian groups in Kurosh's The Theory of Groups. I'm trying to understand the proof of the theorem: Let $B \leqq A$ be abelian groups. If $A/B \cong C$ and $C$ is a free group, ...
1 vote
0 answers
34 views

How to determine $\Gamma_1,\Gamma_2,\Gamma_3,\Gamma_4,\Gamma_5$?

I started learning about Dirichlet Characters. Here is what I learned so far: Definition: Let $m \in \mathbb{N}$. We call a function $\chi:\mathbb{Z} \rightarrow \mathbb{C}$ a Dirichlet Character mod ...
3 votes
0 answers
23 views

Is the localization of a zero-dimensional ring a quotient?

If $R$ is any commutative zero-dimensional ring and $m$ is a maximal ideal, then is $R_m$ always naturally a quotient of $R$? In other words, is the natural map $R\to R_m$ always surjective? I was ...
1 vote
2 answers
582 views

Confusion with application of butterfly lemma in Lang's Algebra

In Lemma 3.3 of Serge Lang's Algebra, the so-called Butterfly Lemma is proved: And then Lang proceeds to prove Schreier's Refinement Theorem (highlight mine): In the proof of Schreier's theorem, ...
2 votes
1 answer
41 views

Factoring polynomials over rings and fields

I would like to determine the factors of $X^N - 1$ over different integer rings, such as $$ \mathbb{Z}[X]\ \mbox{and}\ \mathbb{Z}_p[X]\quad \mbox{for}\quad prime\,\, p $$ In particular, I am ...
3 votes
0 answers
659 views

prove that a conjugate of a glide reflection is a glide reflection

This question has an answer elsewhere, a conjugate of a glide reflection by any isometry of the plane is again a glide reflection, but they use results which were not mentioned in the book. The ...
2 votes
2 answers
87 views

Irreducible factors of $X^n -1$ in $\mathbb{F}_q[X]$

I am stuck trying to prove the following corollary: Let $f = X^n -1 \in \mathbb{F}_q[X]$ with $gcd(q,n)=1$. Let k = order of q mod n. Then the degree of every irreducible factor of $f$ divides k. It ...
2 votes
1 answer
50 views

Attaching an element to a ring $R = \mathbb{Z}/(p^{k}\mathbb{Z})$, assuming it is not in $R$

Let $R = \mathbb{Z}/(p^{k}\mathbb{Z})$, where $p$ be any prime number, and $k > 1$ be any integer. Now let us consider an equation $x^r = p$ in $R$ and $\pi$ be the root of this equation, where $r \...
0 votes
0 answers
25 views

Profinite completion of profinite groups

I was trying to prove that $\mathbb{R}/\mathbb{Z}$ cannot be a galoisgroup for any extension. My plan for this was to show that $\mathbb{R}$ is a divisible group and that every quotient of a divisible ...
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0 answers
50 views

Alternative solution to showing that $\langle x^2 +1, y\rangle$ is a maximal ideal and its possible generalization?

The following is taken from the text An Introduction to Grobner Bases by: Ralf Froberg, and the following Notes: $\langle x^2 +1, y\rangle$ is maximal, pg.3 Question (5a) Background Notation 1: $\...
4 votes
1 answer
69 views

Galois group of $\mathbb{C}(t)$ over $\mathbb{C}(t+t^{-1})$

I was asked to determine the Galois group for the following extensions: first $\mathbb{C}(t+t^{-1})\subset \mathbb{C}(t)$ and then $\mathbb{C}(t^n+t^{-n})\subset \mathbb{C}(t)$ for a certain $n \in \...
0 votes
0 answers
26 views

What do the words "descends" and "Induced" mean in the following quoted passage?

The following is taken from pg 4 section 6.1 of the following notes Background $\quad$ We start by observing that a ring homomorphism descends to quotient rings whenever the image of an ideal on the ...
2 votes
2 answers
817 views

Localization of a module at a prime ideal and local behavior.

Let $M$ be an $R$-module ($R$ commutative with unity) and suppose that given $\mathfrak{p}\subset R$ a prime ideal we have $M_{\mathfrak{p}}\cong R_{\mathfrak{p}}^n$. Is then true that we can find an ...
1 vote
1 answer
28 views

$Out(F_n)$ has a free abelian subgroup of rank $2n-3$

Proposition 9.5.4 The group $Out(F_n)$ has a free abelian subgroup of rank $2n-3$. This is a proposition of the book "Topological Methotods in Group Theory" by R. Geoghegan. In the proof he ...
0 votes
1 answer
128 views
+50

Subgroup of the braid group $\mathbb{B}_n$ generated by $\sigma_1,\dots,\sigma_{n-2},\sigma_{n-1}^2$

I want to prove the following. Let $\mathbb{B}_n:= \left\langle \sigma_1,\ldots,\sigma_{n-1} \mid [\sigma_i,\sigma_{j}]=1,\ j-i>1,\ \sigma_i\cdot\sigma_{i+1}\cdot\sigma_i= \sigma_{i+1}\cdot\...
3 votes
2 answers
93 views

Is $\sqrt{3}$ in the field of 7-adic numbers $\mathbb{Q}_7$?

I am trying to solve the following question: Show that $\sqrt{3} \notin \mathbb{Q}_7$, where $\mathbb{Q}_p$ means the field of $p$-adic numbers. My approach was based on examining the quadratic ...
1 vote
1 answer
75 views

Determine the degree of a root of $X^p-X-\alpha$ (Serge Lang Algebra exercise VI.29)

Let $K$ be a cyclic extension of a field $F$, with Galois group $G=\langle \sigma \rangle$ and assume that $\operatorname{char}F=p$ and that $[K:F]=p^{m-1}$ for some $m>1$. Let $\beta$ be an ...
3 votes
0 answers
30 views

Are there counterexamples of "dividing each other implies association" on a commutative ring but not integral domain? [duplicate]

I am reading about "Fraction on Commutative Ring". On the textbook a proposition states that Let $R$ be a domain and $a,b\in R$. If $a\mid b$ and $b\mid a$, then there exists a unit $u$ ...
0 votes
0 answers
27 views

What is wrong in this proof and where is it needed that $R$ is an integral doamin? [duplicate]

I wanted to prove the following theorem myself: Let $R$ be an integral domain and $p \in R$. If $(p)$ is a maximal ideal, Then $p$ is a prime element. My attempt: Since $(p)$ is maximal then there ...
14 votes
3 answers
3k views

Example of a non-Noetherian complete local ring

I was looking for an example of a non-Noetherian complete local commutative ring with $1$. I would appreciate if anyone can point to a reference.
8 votes
4 answers
937 views

Is the endomorphism ring of a module over a non-commutative ring always non-commutative?

I was reading the "Undergraduate Commutative Algebra". It formalises the definition of Module. Consider M, an A-module where A is a ring. It defines $\mu_f : M \to M$ for the map $m \mapsto fm $, ...
1 vote
1 answer
53 views

$\operatorname{End}_S(M \otimes_R S) \cong \operatorname{End}_R(M) \otimes_R S$?

Let $R$ and $S$ be commutative rings with unity. Let $f: R \to S$ be a ring homomorphism. Let $M$ be an $R$-module. Do we have $$ \operatorname{End}_S(M \otimes_R S) \cong \operatorname{End}_R(M) \...
2 votes
4 answers
214 views

For a transitive and faithful group action the stabilizers are not normal

Let $G$ be a finite group that acts on a set $X$ faithfully and transitively. Additionally we have $|G| > |X| > 1$. Show that there is no $x\in X$ such that its stabilizer is a normal subgroup ...
2 votes
0 answers
43 views

When does the eliminant (resultant) from three variables into two variables vanish?

Definition. Consider a ring $\mathcal{R}$ and polynomials $P,Q\in \mathcal{R}[x]$. We define the eliminant $\mathrm{Elm}(P,Q)$ of $P$ and $Q$ by the determinant of their Sylvester matrix. If $P(x)=\...
0 votes
1 answer
68 views

Questions about a lemma on prime ideals

The following is taken from the text An Introduction to Grobner Bases by: Ralf Froberg Background Definition 63 (Prime Ideal).: An ideal $\mathfrak{p}\neq R$ in a ring $R$ is a prime ideal if $rs\in ...
1 vote
0 answers
36 views

Are determinantal ideals Cohen-Macaulay?

The ideal generated by all the $t$-minors of an $m\times n$ matrix $X$, like $$ X=\begin{pmatrix} X_{11} & X_{12} & \dots & X_{1n}\\ X_{21} & \ X_{22} & \dots & X_{2n}\\ \...
2 votes
0 answers
61 views

Comparing mathematical objects by the "rigidity" of their definitions

A loose interest of mine recently has been ordering mathematical objects by how "combinatorial" their study is, in broad terms. I consider the study of a mathematical object more “...
0 votes
0 answers
19 views

Does the constant term of the minimal polynomial of an algebraic function also have no univariate factor if the algebraic function has none?

Each algebraic function is defined by an irreducible algebraic equation or its minimal polynomial, respectively. My question is: Let ${}^{-}$ denote the algebraic closure, $\pmb{n\in\mathbb{N}_{>1}}...
0 votes
3 answers
91 views

$\gcd(\text{multiples}(a,b))= \text{lcm}(a, b)$?

I am exploring whether the following assertion holds true in integral domains: $\gcd(\text{multiples}(a,b))= \text{lcm}(a, b)$. Let us make this formal below. Consider two elements $a$ and $b$ in an ...
1 vote
1 answer
50 views

...if $\mathfrak{a}\subset\cup_{i=1}^{s}\mathfrak{p_i}$, then $\mathfrak{a_1}\subset \mathfrak{p_i}$ for some $i$

The following is taken from the text An Introduction to Grobner Bases by: Ralf Froberg Background Definition 63 (Prime Ideal).: An ideal $\mathfrak{p}\neq R$ in a ring $R$ is a prime ideal if $rs\in ...
1 vote
1 answer
414 views

Dummit and Foote 10.3.18 (Decomposition theorem for modules)

Let $R$ be PID and $M$ an $R$-Module. Suppose ${\rm Ann}_R(M)=(a)$. Let $a=p_1^{e_1}p_2^{e_2}...p_k^{e_k}$ be a factorization of $a$ into primes. Define $M_i=\{m \in M\ |\ p_i^{e_i}m=0\}.$ Show $M=\...
0 votes
0 answers
27 views

Algebraic simplification of nested radicals and fractions in general?

Let ${}^{-}$ denote the algebraic closure, $n\in\mathbb{N}_+$, $\mathbb{K}\in\{\mathbb{Q},\mathbb{C}\}$, $A(x_1,...,x_n)\in\overline{\mathbb{K}(x_1,...,x_n)}$ (means a rational or irrational $\mathbb{...
0 votes
0 answers
35 views

Correspondence theorem and Quotient ring isomorphism

The following is taken from Abstract Algebra A Comprehensive Introduction by: Lawerence and Zorzitto. Background The ideal in a commutative ring $R$ generated by eleements $a_1,\dots, a_n$ is denoted ...
4 votes
2 answers
374 views

The geometric action of an orthogonal $3 \times 3$ matrix with determinant $-1$.

I am trying to prove that the action of an orthogonal $3\times 3$ matrix with determinant $-1$ is a reflection about an eigenvector in one of the matrix's eigenspace, but I am a little lost. Problem ...
0 votes
2 answers
33 views

Relationship Between Subgroups of Abelian Groups & Ideals/Rings.

Clarification. I am currently reading from Dummit and Foote. Given $R$-module $M,$ we require $(1)$ $R$ is unital, and $(2)$ $1\cdot x=x$ for all $x\in M.$ When discussing rings $R,$ for the purposes ...
0 votes
0 answers
32 views

Is $B$ flat as an $A$-module?

Suppose $A$ is an integral closed domain, and its quotient field is $K$. Suppose $L$ is a finite separable extension of $K$, and $B$ is the integral closure of $A$ in $L$. Is $B$ flat as an $A$-...
1 vote
1 answer
410 views

Problem on a Bilinear pairing

This is a problem from hatcher's that I am trying to solve. Show that a nonsingular symmetric or skew-symmetric bilinear pairing over a field $F$, of the form $F^n\times F^n\rightarrow F$, cannot be ...
3 votes
0 answers
83 views

The condition of compactness of subgroup of infinite Galois group

Question: Let $E$ be Galois over $F$, with Galois group $G$. $H$ is a subgroup of $G$. Prove that $H$ is dense in $G$ if and only if for any subfield $K$ of $E$ containing $F$ and $K$ finite and ...
1 vote
0 answers
40 views

Conceptual definition of the Auslander-Reiten translate

In homological algebra, we learn to differentiate between The conceptual definition. A computation, which is done by choosing efficient resolutions. The only definition I've seen of the Auslander-...
3 votes
0 answers
69 views

Estimation of the eigenvalue of a matrix

Let $A_n$ be a $n×n$ matrix with entries $a_{ij}=n-|i-j|$, let $\lambda_n$ be the largest eigenvalue of $A_n$, find $\lim_{n \to \infty} \frac{\lambda_n }{n^2}$ I find it hard to compute the ...
2 votes
2 answers
288 views

Free group on a set $X$ is generated by $X$, why?

The definition of free group given in class was: Given a non-empty set $X$ and an inclusion $i:X\rightarrow F$. We say that $F$ is a free group on $X$ if, given any group $G$ and a set map $\varphi:X\...
0 votes
0 answers
17 views

When a quotient of $\mathbb Z_p[[x,y]]$ is a regular local ring

Consider the power series in two variables with p-adic coefficients. Namely consider $\mathbb Z_p[[x,y]]$. Moreover let $c\in\mathbb Z_p$ and construct the quotient: $$ W=\mathbb Z_p[[x,y]]/(xy-c) $$ ...
0 votes
0 answers
23 views

A problem of determining whether a power series belongs to $\mathbb{C}(u)$

I am reading a paper "Drinfeld coproduct, quantum fusion tensor category and applications" and I have a probelm. Here is the arxiv:Drinfeld coproduct, quantum fusion tensor category and ...
0 votes
1 answer
57 views

Why $P(y/x) = 0$?

In Stack Project Commutative Algebra 119.2, $(R,m)$ is a local Noetherian ring. In one case of the proof, $x \in m$ is a nonzerodivisor, $y \in R$ and $ym \subset xm$. Then we have a map $$ \varphi: m ...
0 votes
0 answers
36 views

Question about finding generators of the kernel for a substitution maps in rings.

The following is taken from Abstract Algebra A Comprehensive Introduction by: Lawerence and Zorzitto. Background Exercsie 6: Find generators for the kernel of each of the following substitution maps: ...
4 votes
2 answers
617 views

Showing tensor product of coalgebras is a coalgebra.

Let $(C, \Delta, \epsilon)$ and $(C',\Delta', \epsilon')$ be two coalgebras over the field $k$. I'm trying to show that $C \otimes C'$ is a coalgebra for the comultiplication $$\overline{\Delta}:=(id_{...
15 votes
2 answers
13k views

Simple module is isomorphic to $R/M$ where $M$ is a maximal ideal

In Michael Artin's Algebra textbook page 484 Chapter 12 Exercise 1.6: A module is called simple if it is not the zero module and if it has no proper submodule. (a) Prove that any simple module is ...
0 votes
0 answers
31 views

Proof that the order of elements stays the same under Frobenius endomorphism

Let $E: Y^2=X^3+Ax+B$ be an elliptic curve, defined over $\mathbb{F}_p$ where $p$ is a prime. Define: $$\phi: E(\bar{\mathbb{F}}) \rightarrow E(\bar{\mathbb{F}})$$ by $$\phi(P) = \left\{ \begin{array}...

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