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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Can this be done using Sylow theorems? [duplicate]

Let $p$ and $q$ be distinct primes.Suppose that $H$ is a proper subset of the integers and $H $is a group under addition that contains exactly three elements of the set {$p,p+q,pq,p^q,q^p$}.Determine ...
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3answers
67 views

On a property of generating set for groups

Consider the following paragraph from Lang's Algebra: My question is about converse of one statement, which I was not able to prove. Question: If $f(S)$ do not generates $F$, then can we find ...
3
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1answer
72 views

Dual space of polynomial algebra

Let $k$ be an infinite field and let's consider the ring $R=k[x_1,\dots,x_n]$. This ring has a structure of $k$-vector space (or a $k$-algebra). I am interested to know about the structure of the ...
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1answer
40 views

Is the universal enveloping algebra functor exact?

The universal enveloping algebra is a functor from Lie algebras to unital associative algebras, and is left adjoint to the functor which sends a unital associative algebra to a Lie algebra with ...
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2answers
107 views

Is $x^5 + x^3 + 1$ irreducible in $\mathbb{F}_{32}$ and $\mathbb{F}_8$?

Problem: Is $f(x) = x^5 + x^3 + 1$ irreducible in $\mathbb{F}_{32}$ and $\mathbb{F}_8$? My thought: $f(x)$ is irreducible in $\mathbb{F}_2$ and has degree $5$. So we can conclude that $\mathbb{F}_{...
3
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1answer
25 views

Field and ideal notation: double bracks/parens vs single brackets/parens

I'm reading some notes that has the following denotation: the set of formal power-series with coefficients in $\mathbb{F}_p$ is denoted by $\mathbb{F}_p[[t]]$. the fraction field, $\operatorname{...
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1answer
58 views

Need help in understanding the proof of “If $ \vert G \vert$=60 and $ G $ has more than one Sylow 5-subgroup, then $ G $ is simple.”

This proof is from Dummit & Foote text. Suppose by way of contradiction that $\vert G \vert=60$ and $n_5$>1 but that there exists $H$ a normal subgroup of $G$ with $H$ $\neq$ $1$ or $G$. By Sylow'...
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3answers
69 views

Are Pandemic chain reactions confluent? (vertex spills weight to neighbors at threshold, once)

Are resolutions of chain reactions order-independent in the board game Pandemic? More formally: You're given an undirected graph $G = (V, E)$ and a vertex weight $w \colon V \to \{0, \ldots, 3\}$. ...
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0answers
33 views

What kind of map is the inversion map on groups?

Let $G$ and $H$ be groups. Suppose $\phi : G \rightarrow H$ is a map such that $\phi(g g') = \phi(g') \phi(g)$ for all $g,g' \in G$. What is the name for such a map? For example, if $H = G$, then $\...
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2answers
83 views

Commutativity and $(a + b)^2$

I read "note that if a and b are commutative, $(a + b)^2 = a^2 + 2ab + b^2$". Could someone explain how we need commutativity for this to happen?
2
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1answer
34 views

irreducible unitary reflection group

Let $G$ be a finite irreducible unitary reflection group (i.e. without G-invariant subspaces). Given orthonomal basis, we have that $g_1 \in GL(V)$ commutes with every element of $G$. It is said that ...
2
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1answer
41 views

Determining a group given some elements

Say we have a group G and we know some of the elements (but not all). How does one determine the order and list all the elements of the group in an intuitive way? In this case G is the smallest ...
2
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1answer
50 views

A doubt about the correspondence theorem.

Let $f$ be a ring homomorphism from $R$ onto $R_1$. Then there is a one one correspondence between the set of all ideals of $R_1$ and the set of all ideals of $R$ that contain the kernel. Now what ...
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1answer
57 views

What can I say about the maps $\text{Spec} (A / \mathfrak{a}) \to \text{Spec} (A)$ geometrically?

I was curious whether there is a general approach to say something about the geometric interpretation of the maps $\text{Spec}(A / \mathfrak{a}) \to \text{Spec}(A)$ for a commutative ring with unity $...
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2answers
35 views

Order of factor group

Question: Determine the order of $(\mathbb{Z} \times \mathbb{Z})/ \left<(4,2)\right>$. Is the group cyclic? I want to first apologize for the way this post is written. I'm on the road and ...
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0answers
27 views

Elementary literature on Group theoretic Power Diophantine Equation

I am looking for an elementary books/pdf notes on group theory related to Power Diophantine Equation. I have read elementary group theory. Please advise some books/pdf notes. Also, it would be ...
2
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3answers
89 views

Subgroup of Order $n^2-1$ in Symmetric Group $S_n$ when $n=5, 11, 71$

$G$ is a symmetric subgroup of symmetric group $S_n$ acting on $n$ objects where $n=5, 11, 71$ and order of $G$ is $n^2-1$. Question: Does $G$ (as defined above for $n=5, 11, 71$) exist? How can I ...
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2answers
42 views

Questions on inverse limits

Let $(S_n,\pi_n)$ be an inverse sequence, where $S_n=\mathbb N$ for each $n$. If each $\pi_n$ is identity, then the inverse limit, $\varprojlim S_n$, is (bijective to) $\mathbb N$. My first question ...
0
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1answer
26 views

About Structure of Free Algebra over $K$

In MIT Course No. $18.712$, Associative Algebra $A$ is defined as a vector space over a field $K$ with a bilinear associative map $A \times A \to A$, $(a,b) \to ab$. Then some examples are given, ...
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1answer
100 views

What are some good books and materials for studying rings and fields theory? [closed]

I will very soon be introduced to the subject. I have heard this is one of the most important part of undergraduate algebra. I want to develop clear understanding in it from the beginning. I have ...
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1answer
41 views

Universal property of generating set for vector space

Let $V$ be a vector space over $F$, and $S$ a non-empty subset of $V$. We say that $S$ generates $V$ if every $v\in V$ can be written as finite $F$-linear combination of elements of $S$. I want to ...
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1answer
44 views

Does $\text{Gal}_{\mathbb{Q}}(\overline{\mathbb{Q}})$ act on $\overline{\mathbb{Q}}$ with an infinite orbit?

Does $\text{Gal}_{\mathbb{Q}}(\overline{\mathbb{Q}})$ act on $\overline{\mathbb{Q}}$ with an infinite orbit? I know the definition of orbit and I know that $\sigma$ in Galois group changes roots. I ...
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2answers
60 views

What does $V^*$ means?

What does it mean to have an "$f \in V^*$" in terms of a transformation? The chapter in the book it is in is about dimensions in vector spaces.
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3answers
54 views

$\ker \phi = (a_1, …, a_n)$ for a ring homomorphism $\phi: R[x_1, …, x_n] \to R$

Let $R$ be a commutative ring, $a_1, ..., a_n$ its elements and $\phi: R[x_1, ..., x_n] \to R$ defined by $ \phi(f(x_1, ..., x_n)) = f(a_1, ... ,a_n)$ a ring homomorphism. Prove: $\ker \phi = (...
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2answers
52 views

When is $\overline{K}/K$ a Galois extension of $K$?

When is $\overline{K}/K$ a Galois extension of $K$, where $\overline{K}$ stands for the algebraic closure of $K$? I have the following three extensions: $\overline{\mathbb{Q}}/\mathbb{Q}$,$\...
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2answers
25 views

Prove that $\frac{R/ \ker \phi}{(\ker \phi + J)/ \ker \phi} \cong \frac{S}{\phi(J)}$

Let $\phi: R \to S$ be a surjective homomorphism. Prove that $\frac{R/ \ker \phi}{(\ker \phi + J)/ \ker \phi} \cong \frac{S}{\phi(J)}$ for an ideal $J$ of $R.$ Obviously, $S \cong R/ \ker \phi$(first ...
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0answers
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Let $k$ be a field of characteristic $\neq 2$. Then $x^6-xy+y^6$ is irreducible in $k[x,y]$. [duplicate]

There is no obvious way to apply Eisenstein's criterion; and if I assume by contradiction that $x^6-xy+y^6=f(x,y)g(x,y)$, f with homogeneous degree $\leq 3$. Then I have that $f(x,y)=\sum_{i,j} a_{ij}...
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1answer
53 views

Is a fiber product of flat morphisms flat?

Suppose we have morphisms of schemes $f : X\rightarrow S$ and $g : Y\rightarrow S$, and a morphism $Z\rightarrow X\times_S Y$ such that the induced morphisms $Z\rightarrow X, Z\rightarrow Y$ are flat. ...
3
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0answers
51 views

Are the elements of a module also called vectors?

Are the elements of a module also called vectors? Or if someone says 'vector', are they talking only about a vector space? If no context is given, are there some standard assumptions?
3
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5answers
101 views

Is it true that $\mathbb{Q}(\sqrt{2},e^{2i\pi/3}) = \mathbb{Q}(\sqrt{2}+e^{2i\pi/3})$?

Is it true that $\mathbb{Q}(\sqrt{2},e^{2i\pi/3}) = \mathbb{Q}(\sqrt{2}+e^{2i\pi/3})$? I know that $[\mathbb{Q}(\sqrt{2},e^{2i\pi/3}):\mathbb{Q}]=2\times2=4$. By using WolframAlpha (cheating), I know ...
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1answer
30 views

Counting subgroups of free product of copies of $\mathbb{Z}$ with certain index

For a natural number $n$, let $Z_n=\mathbb{Z} \ast \cdots \ast \mathbb{Z}$ denote the free product of $n$ copies of the integers. Let $m$ be a further integer. $\textit{Question:}$ Is there a ...
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0answers
24 views

Relation between permutation group and algebraic equation. [duplicate]

What kind of relation do algebraic equeation and permutation group have? For example, $Z^n -1=0$ is related to a cyclic group $C_n$. Is there anything else in this kind problem? I have read about ...
2
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1answer
33 views

Permutation of Disjoint Sets of a Symmetric Group

Problem Description: Consider a symmetric group $S_n$ acting on $n$ objects. We partition $S_n$ into two sets $A, B$ such that $A \cap B= \emptyset$ and $A \cup B = S_n$. In other words, $S_n$ is ...
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0answers
38 views

Idempotents central or not?

Let $R$ be a nil-clean ring with unity such that $R/J(R)$ is reduced, where $J(R)$ is the Jacobson radical of $R$. Is it true that $R$ is abelian, i.e. the idempotents are central? (By nil-clean I ...
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0answers
23 views

How does restriction of scalars interact with tensor products?

Say that we have a morphism of commutative rings $f: R \to S$. Does the restriction of scalars functor $f^*: S \text{Mod} \to R \text{Mod}$ commute with tensor products? In other words, I would like ...
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0answers
20 views

how do we get the given diophantine equation and it's value

\label{thm:3} The only squares of the form $${\overline{aa \ldots ab \ldots b}}_{(10)}$$ in decimal representation are the trivial infinite families $10^{2i},\; 4\cdot 10^{2i}$, $9\cdot 10^{2i}$ with $...
6
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2answers
198 views

Is the projective line minus one point always isomorphic to the affine space?

I'm thinking about the following problem: If I take a general point $p \in \mathbb{P}^1$ out of the projective line, is $\mathbb{P}^1 - \{ p \}$ isomorphic to the affine space $\mathbb{A}^1$? I ask ...
4
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2answers
170 views

Eigenvalues of matrices over finite fields

I apologize in advance if this is trivial, but I am a bit confused here. So consider the finite field $\mathbb{F}_{p^d}$ over the prime field $\mathbb{F}_p$. Wecan associate with every element $\...
3
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2answers
104 views

How to write the commutator subgroup in terms of the generators of the group?

Let $G=\langle\ S\ |\ R\ \rangle$ be a finitely presented group. The commutator subgroup of $G$ is the group generated by $\{[a,b]\ |\ a,b\in G\}$ and is denoted by $[G,G]$, where $[a,b]=aba^{-1}b^{-1}...
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Intuitive reasons of ring modulo maximal ideal or prime ideal

Are there any intuitive reasons that can help us remember that $R/I$ is a field iff $I$ is a maximal ideal; $R/I$ is an integral domain iff $I$ is a prime ideal? (I can understand the proof, but have ...
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theorem 2 of perfect powers with all equal digits but one

Can someone help me understand what happened to this equation from the paper entitled Perfect Powers With All Equal digits but one...I don't understand the part when it lets $a$ and $c$ not equal to ...
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0answers
30 views

Proving that if T∈Hom[V,W] has null space X∩Y, then T[X+Y]=T[X]⊕T[Y]

Here's my progress: If X and Y are subspaces of a finite-dimensional vector-space, then d(X+Y)+d(X∩Y)=d(X)+d(Y) , where d(A) is the dimension of A. Then, d(X∩Y)=d(X)+d(Y)-d(X+Y). But X∩Y is ...
2
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3answers
91 views

If $G/N\cong H$ then $G=NH$?

I am curious if $G/N\cong H$ then $G=NH$? ($N$ is a normal subgroup of $G$, and $H$ is a subgroup of $G$.) With this setup, we get that $NH$ is a subgroup of $G$ so $NH\subset G$. I am not sure ...
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0answers
36 views

Methods of determining irreducibility of a polynomial in a large finite field

Given a finite field $\mathbb{F}_p$ and some polynomial $f(x)\in\mathbb{F}_p [x]$. What are some of the methods of determining the irreducibility of $f(x)$? I feel like there are many theorems that we ...
2
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0answers
32 views

Maximal unramified intermediate extension of DVRs

Let $A\rightarrow B$ be a finite tamely ramified extension of discrete valuation rings. Does there exist a DVR $C$ such that $A\subseteq C\subseteq B$ with $C$ unramified over $A$ and $B$ totally ...
2
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2answers
85 views

Show that $K(a_1, \dots , a_n)=K[a_1, \dots , a_n]$

Let $L/K$ be a field extension and $a_1, \dots a_n\in L$, such that $a_1$ is algebraic over $K$, $a_2$ is algebraic over $K(a_1)$ and in general, $a_i$ is algebraic over $K(a_1, \dots , a_{i-1})$ for $...
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4answers
100 views

Why should $b$ groups of $a$ apples be the same as $a$ groups of $b$ apples?

Why should $b$ groups of $a$ apples be the same as $a$ groups of $b$ apples? We where taught this so it seems rather trivial but the more I think about it the more I feel that it is not. I'm trying ...
3
votes
1answer
55 views

Is the set of hyperreal numbers a quotient ring?

It is easy to see that the set of real sequences $\mathbb{R}^{\mathbb{N}}$ is a ring. It suffices to define, for all $r,s\in\mathbb{R}^{\mathbb{N}}$, the operations $r\oplus s =(r_n+s_n)_{n\in\mathbb{...
2
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3answers
60 views

Alternate solutions to algebraic equations? [closed]

Most people have probably seen silly tricks in mathematics where people make the take simple steps and end up proving 1=0 or some similarly absurd result. As an example look at the (object/number/...?)...
1
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3answers
55 views

Integral closure of Gaussian Integers

I am considering $\mathbb{Z}[i]\subset\mathbb{Q}(i)$ Now I have a short note here that says that there are elements of $\mathbb{Q}(i)$ which are not integral over $\mathbb{Z}[i]$ 'because $\mathbb{Q}(...