# Tagged Questions

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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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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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### 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 ...
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### 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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### 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 ...
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### 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 ...
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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 $... 4answers 100 views ### Why should$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$apples? We where taught this so it seems rather trivial but the more I think about it the more I feel that it is not. I'm trying ... 1answer 55 views ### Is the set of hyperreal numbers a quotient ring? It is easy to see that the set of real sequences$\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{...
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}(...