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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Smoothness in cyclotomic versus complex fields?

Say we have a polynomial in a cyclotomic field; in particular, an n-th cyclotomic field, where n is the order of the polynomial's symmetry group. If we know the polynomial is smooth over this field, ...
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32 views

Is there any group-like structure that doesn't have an identity, but has (non-equal) left and right identities?

Is there any group-like structure that doesn't have an identity, but has a (non-equal) left and right identities?
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Representation of $(\mathbb{Z}_{\frac{*}{5}},_{\times5})$ using Cayley table [on hold]

Could someone give a hint how to represent group $(\mathbb{Z}_{\frac{*}{5}},_{\times5})$ using Cayley table? Thanks for replies.
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35 views

Conjecture on a graded ring

Consider $B^{(n)}=\mathbb{F}_2[X_1,\dots,X_n]/(X_1^2,\dots,X_n^2)=\bigoplus_{i=0}^nB_i^{(n)}$, where $B_i^{(n)}$ is the space of homogeneous elements of degree $i$. Notice that ...
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1answer
44 views

Cancel an element that's not a unit

I was going through the proof of "every PID is a UFD" in Serge Lang's Algebra-book and something confused me. When it comes to proving the uniqueness of the factorization, he writes: "Suppose $a$ ...
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Isomorphism Between Splitting Fields

Let $f \in F[x]$ be irreducible and split in $E/F$. Suppose that $F \cong F'$ via $\phi$, and let $f' \in F'[x]$ be polynomial obtained by applying $\phi$ to the coefficients of $f$. It can be shown ...
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1answer
23 views

What is the size of Range?

Suppose d=gcd$(a,n)$ where $a, n \in \mathbb{Z}, n>0$ and $f_a: \mathbb{Z_n} \to \mathbb{Z_n} \\ \qquad x \to ax\texttt{ mod } n$ The size of Domain is evident and for the size of Range my ...
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Milnor patching for general modules

The Milnor patching theorem for projective modules is the following statement. Given a pullback diagram of rings $$ \begin{array}{} R & \xrightarrow{f_2} & R_2 \\ \downarrow{f_1} & ...
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Derivations of algebra matrices?

I see on internet that all derivations of algebra of matrices $M_n(\mathbb{R})$ with respect to it commutator or matrices multiplication, are inner. I do not know how to prove this fact. Any hint are ...
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27 views

Expressing polynomial as linear combinaion

I found these questions in Adams Introduction to Groebner bases Let $f=x^6-1$ and $g=x^4+2x^3+2x^2-2x-3$. Let $I=\langle f,g\rangle$. Calculate the polynomial that generates $I$ alone. After a ...
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1answer
46 views

Maximal Ideals of $\mathbb C[x, y]$

I recently learnt that the maximal ideals of $\mathbb C[x, y]$ are precisely the ones of the form $(x-a, y-b)$ for some $a, b\in \mathbb C$. I am unable to prove it. So I considered an easier ...
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Intuition on $S_4/K_4\cong S_3$ [duplicate]

This is a question I was pondering on my way to class, and may not have an answer. Take the normal copy of $K_4$ in $S_4$ (The group $K_4 \cong\{e,(12)(34),(13)(24),(14)(23)\}$, not the non-normal ...
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1answer
18 views

Induction and Compact induction of representations

Let $H \leq G$ be a subgroup of a finite group, $G.$ Suppose $(\sigma, W)$ is a representation of $H.$ Then we know that $Ind_H^G \sigma $ and $ind_H^G \sigma $ are isomorphic, where $$Ind_H^G ...
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1answer
38 views

Localization of a coherent module is coherent

I would like to prove that the coherence of a sheaf on a scheme is affine-local. For this, it is necessary to prove that for a coherent $A$-module $M$, its localization $M_{f}$ at $f\in A$ is a ...
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132 views

Is there any neat way to show $\phi$ is a homomorphism?

In Michael Artin's Algebra (chapter 2, page 50, example 2.5.13) the author illustrates a homomorphism from $S_4$ (all permutations of indices $(1,2,3,4)$) to $S_3$ (all permutations of indices ...
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3answers
31 views

Number of left cosets equals number of right cosets?

So I've been working on abstract algebra out of John B. Fraleigh's 3rd edition text. In the exercises of chapter 11, I came upon a question which I cannot even begin to solve. "Show that there are ...
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1answer
63 views

Why would we a priori expect $V(I)$ to satisfy axioms to define the closed sets for a topology on $\text{Proj}(S)$?

The topological space $\text{Proj}(S)$ has the underlying set$$\text{Proj}(S) = \{\mathfrak{p} \text{ a homogeneous prime such that }S_+ \not\subseteq \mathfrak{p}\},$$and the closed sets are the loci ...
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20 views

For a root system, why does $\beta\in\Delta_+\setminus\{\alpha_i\}$ imply $(\beta+\mathbb{Z}\alpha_i)\cap\Delta\subset\Delta_+$?

Let $\mathfrak{g}(A)$ be a Kac-Moody algebra for a matrix $A$, with root basis $\{\alpha_1,\dots,\alpha_n\}$. There is a remark on the bottom of page 6 of Kac's Infinite Dimensional Lie Algebras ...
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Why is $\mathfrak{g}(A)=\mathfrak{g}'(A)$ iff $\det(A)\neq 0$?

In many sources (Victor Kac, Zhexian Wan, etc.), it's stated as a remark that if $\mathfrak{g}(A)$ is the Kac-Moody algebra of a generalized Cartan matrix $A$, then $\mathfrak{g}(A)=\mathfrak{g}'(A)$, ...
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2answers
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How do we conclude that the relation is equal to $1$ ?

We have a curve of the form $$s^2-\alpha t^2=1 \tag 1$$ (in $ts$-coordinates). If $(a,b)$ and $(c,d)$ are points of $(1)$, then $(ac+\alpha cd, ad+bc)$ is point of $(1)$. $(a,b)$ is a point of ...
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1answer
35 views

$R$ is a unique factorization domain $\iff$ every prime minimal over a principal ideal is also principal

I'm trying to show that a ring $R$ is a unique factorization domain $\iff$ every prime minimal over a principal ideal is also principal. I think the idea is to use the principal ideal theorem of ...
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2answers
63 views

Commutative binary operations on $\Bbb C$ that distribute over both multiplication and addition

Does there exist a non-trivial commutative binary operation on $\Bbb C$ that distributes over both multiplication and addition? In other words, if our operation is denoted by $\odot$, then I want the ...
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Show that every algebraic field extension has a normal closure

I am trying to show that every algebraic extension of fields has a normal closure. Let $L/K$ be an algebraic extension. First suppose that the extension is finite. So $L=K(\alpha_1,...,\alpha_n)$ ...
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Is there a polynomial $p$ such that it is bijective and $ p: \mathbb{Q}^n \rightarrow \mathbb{Q}$ for $ n>1$ ??

Let us define a polynomial $p$ defined as follow $$p: \mathbb{Q}^n \rightarrow \mathbb{Q}.$$ I acrossed this question in mind. Is there a polynomial $p$ such that it is bijective and $p: ...
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2answers
63 views

In how many ways can a group act on a set?

$Z_{5}$ is the group to act on the set $\{ 1,2,3,4,5\}$. In how many ways is that possible $?$ Now $0$ will give the identity map. $1$ will give a bijection in $5!$ ways so ...
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16 views

Set of splittings of a short exact sequence of modules

For a short exact sequence of modules $$0 \rightarrow M \rightarrow N \rightarrow R \rightarrow 0$$ with $\phi : M \rightarrow N$ and $\psi : N \rightarrow R$, a splitting is an isomorphism $\xi : ...
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1answer
51 views

Is this group of matrices a $p$-group?

Let $R$ be a discrete valuation ring with the maximal ideal $\mathfrak{m}=(\pi)$ and residue field $k$ of positive characteristic $p$. Now consider $\mathrm{M}_n(\pi^iR/\pi^{i+1}R)$, $n\times n$ ...
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exercise questions [on hold]

Teacher, would you like to tell me link to get pdf file of solution manual for Gallian'abstract algebra.
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1answer
79 views

Difficult to read about different subjects simultaneously, should I leave one for now? [on hold]

I learn math by reading books. Usually I read 3 books (about 3 different subjects) simultaneously and switch focus every couple of days. The books i'm studying right now are Rudin's functional ...
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1answer
37 views

the unit group of an infinite field cannot be cyclic [duplicate]

It is well-known that the unit group of a finite field is a finite cyclic group. But for infinite fields, e.g., $\mathbb{Q}$ or $\mathbb{R}$, the unit groups are not cyclic. I heard this fact in my ...
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0answers
38 views

About some quotient of finitely generated group

Let $G$ be a finitely generated group. Assume there exists $N$ an abelian normal subgroup of $G$, such that $G/N=H$ with $H$ a finite group of order not divisible by a fixed prime number $p$. In ...
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1answer
41 views

Polynomial algebras

What is a polynomial algebra? I cannot find any definition of this concept. Is it a set of n-ary polynomial symbols defined over an algebra, as defined on p.29 of this document? ...
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39 views

Free lattice in three generators

By general results for every set $X$ there is a free bounded lattice $L(X)$ on $X$. I would like to understand the element structure of this lattice. The cases $X=\emptyset$, $X=\{x\}$ and $X=\{x,y\}$ ...
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2answers
64 views

The relationship between subfields and subgroups of a finite field.

I am trying to get my head around the structure $GF(p^n)$ when viewed as a vector space of dimension $n$ over $GF(p)$ (mainly the relationship between the additive and multiplicative structures). I'm ...
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3answers
40 views

order and cycles of perfect shuffle of 52 cards

This is the shuffle: $$1,2,\cdots,52$$ is turned into $$1,27,2,28,\cdots,26,52$$ when I try to write the cycles of this shuffle, I get a LOT of cycles, for example: $$(2\ 3)(27 \ 2)(26\ ...
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4answers
50 views

How to show that the cycle $(2 5) = (2 3) (3 4) (4 5) (4 3) (3 2)$

I generally do not have any problem multiplying cycles, but I've seen on Wikipedia that $$(2 5) = (2 3) (3 4) (4 5) (4 3) (3 2). $$ I started following the path of $2$ on the right: ...
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2answers
28 views

Free product of two algebras and actions of algebras.

Let $A, B$ be two algebras. Suppose that $A$, $B$ acts on $V$. Then we have two maps $$ \delta_1: A \otimes V \to V, \\ \delta_2: B \otimes V \to V, $$ which satisfy the axioms of actions. Do we ...
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0answers
29 views

e solutions of gallian abstract algebra 4th edition [on hold]

teacher, Please help me, I am trying my level best but still not found ebook for solutions of Gallian abstract algebra. But I hope I will get soon. Thanks a lot
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57 views

Proof that the Cyclotomic Polynomial is the Minimal Polynomial for an n-th Primitive Root [on hold]

Let $p(x)$ be a polynomial over $Q[x]$. Prove that if $p(x)$ has as root a 5th primitive root of unity, then all 5th primitive roots are roots of $p(x)$ How Can I prove it? This would mean $p(x)$ is ...
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1answer
30 views

Partition of a group such that an operation can be defined

I'm struggling with Problem 43 of 3.1 of Dummit's algebra book. The problem is: Assume $P=\{A_i\}$ is any partition of $G$ with the property that a "quotient operation" is defined as follows: to ...
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1answer
59 views

Prove that for a group with even order $2k$, there is a subgroup $K$ with order $k$

I'm trying to understand the proof my teacher did: Consider a subgroup $H$ of $G$. If $H$ is not contained in $A_n$, then we can say that there exists at least one permutation in $H$ that is odd ...
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1answer
31 views

Extending a Field Monomorphism

In theorem A3.5 of Ash's book Abstract Algebra: The Basic Graduate Year (page 20 in this pdf), the author set out to prove the following. Let $\sigma: F \rightarrow L$ be a field monomorphism ...
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Binary solutions of multivariate polynomial system in special (factored) form.

In my personal research I've run into a system of multivariate polynomials (with coefficients in a field). I am aware that there is no polynomial time algorithm (in the number of indeterminates) for ...
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1answer
77 views

Permutations of the elements of $\mathbb Z_p$

Note Added by Robert Lewis, 2 August 2015 3:04 PM PST in an attempt to provide background, motivation, and other context for this engaging problem: This problem essentially asks for a method of ...
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27 views

Relation between a group's cardinality and number of subgroups

Why are these following situations not possible? A. An infinite group has finite number of subgroups B. An uncountable group has countable number of subgroups. Any infinite ...
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1answer
35 views

Is $K := \mathbb{Q}(\cos (2\pi / 11))$ a Galois extension over $\mathbb{Q}$?

I believe that it is because $\cos(2\pi / 11) = (\zeta + \zeta^{-1})/2$ where $\zeta = e^{2\pi i/11}$ is a primitive $11$-th root of unity, and so $K$ is a subfield of $\mathbb{Q}(\zeta)$ with ...
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1answer
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What is the fastest, most correct way to solve this simultaneous of two linears?

\begin{eqnarray*} (x+2)/5-((y+2)/4) &=& 2-(x/3) \\ (x+5)/4+((x-y)/5) &=& y+5 \end{eqnarray*} What is the fastest, most correct way to solve this simultaneous of two linears?
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45 views

How to solve conjugation equations in group theory

Given the permutations $(12)(34)$ and $(56)(13)$ find $a$ such that $$a^{-1}xa = y$$ I just realized that I don't know how to solve this exercise. My book don't even give examples of how to solve ...
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1answer
44 views

Associate a discrete valuation ring to a field $k$.

I have a field $k$ of positive characteristic $p$, not necessarily perfect. Can i find a discrete valuation ring that have $k$ as residue field and field of fractions $K$ of characteristic zero? I ...
2
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1answer
43 views

Prove gN in G/N has infinite order.

Let G be a Group and let g in G. Let g have infinite order and suppose that N is a finite normal subgroup of G. Prove that the Element gN in G/N has infinite Order. I started by proving by ...