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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Finite group $G$ satisfies that for all positive integer $m$ number of solutions of $x^m=e$ are at most $m$ is cyclic. [duplicate]

Finite group $G$ satisfies that for all positive integer $m$ number of solutions of $x^m=e$ are at most $m$ is cyclic. It is exercise in A First Course in Abstract Algebra by Fraleigh. The book ...
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Injective map from the Cartesian product of two sylow-p-subgroups into the group.

Let $G$ be a group of order 148. Show that $G$ is not simple. The given solution goes as follows: $148 = 4 × 37$. By Sylow’s theorem, it has at least one subgroup $P$ of order 37. If $P'$ is ...
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Genetic algebras; what should a mathematician know about them?

I recently learned that genetic algebras are a thing, which means there is a link between abstract algebra and genetics. Question: What should I (wearing my mathematician's hat) know about them? ...
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$\mathbb{Q}(\sqrt{23})$ is not a Euclidean number field.

The problem I'm facing is that of the tittle: Problem. Prove that $\mathbb{Q}(\sqrt{23})$ is not a Euclidean number field. Since $23\not\equiv 1\pmod{4}$, it must be shown that ...
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1answer
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Finding a homomorphism from a subset of the fractions to the ring $\mathbb{Z}_p$.

I'm practicing using the First Isomorphism Theorem for rings. Here is a question I got stuck on. Let $p$ be prime and let $T$ be the set of rational numbers (in lowest terms) whose denominators ...
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1answer
36 views

Quotient ring $R/I$, $R= \mathbb{Z}[\sqrt{-10}]$, $I =(\sqrt{-10})$

To show $I =(\sqrt{-10})$ is not prime in $R=\mathbb{Z}[\sqrt{-10}]$, there is a direct method, i.e. show that $2$ and $5$ are not in $I$ but their product $10$ is. However, I struggled to prove the ...
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Let $A$ be a ring, $M''$ and $P$ be $A$-modules and $f:P\rightarrow M''$ a homomorphism…

I need some help with this problem, it seems easy, but I don't get it. Let $A$ be a ring, $M''$ and $P$ be $A$-modules and $f:P\rightarrow M''$ an homomorphism. Suppose that for each $A$-modules $M$ ...
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2answers
53 views

Why is $\mathbb{Q}(\sqrt{2}\sqrt[3]{3}) \subset \mathbb{Q}( \sqrt{2},\sqrt[3]{3})$

Why is $\mathbb{Q}(\sqrt{2}\sqrt[2]{3}) \subset \mathbb{Q}( \sqrt{2},\sqrt[3]{3})$ "obvious"? My book states this as obvious, but then proves the opposite inclusion. I would have thought that ...
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77 views

Transitivity-like Results in Group, Ring, Module, Field and Galois Theory [closed]

I am reading Michael Atiyah and Ian Macdonald's Introduction to Commutative Algebra. On page 28, Proposition 2.16 says: Suppose $A,B$ are rings, $N$ is a finitely generated $B$-module, $B$ is ...
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20 views

Conjugacy of Hall $\pi$-subgroups of a group $G$

A group is called a $C_\pi$-group if there exists a Hall $\pi$-subgroup and any two Hall $\pi$-subgroups are conjugate. Let $G$ be a group such that $N$ is a normal $C_\pi$-subgroup of $G$ and ...
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$Aut(S_n)=Inn(S_n)$ for all natural number n except 6. [duplicate]

Since $G/Z(G)$ is isomorphic to $Inn(G)$ and $Z(Sn)=1$, If $Aut(S_n)=Inn(S_n)$, then $S_n=Aut(S_n)$. How can I show that $Aut(S_n)=Inn(S_n)$ for except $n=6$?
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78 views

Given two algebraic conjugates $\alpha,\beta$ and their minimal polynomial, find a polynomial that vanishes at $\alpha\beta$ in a efficient way

Inspired by this question, I was wondering about the following problem. $\alpha,\beta,\gamma,\ldots$ are the roots of an irreducible polynomial over $\mathbb{Q}$. How to compute the coefficients ...
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40 views

Does charR=0 imply that R is a field? [closed]

What are the implications of charR=0? If D is an ID then charD=0 or p. I don't know if the converse is true.
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1answer
39 views

Number of elements in $U(n)$ with order dividing $n-1$ [closed]

Please suggest a solution to this problem: Let $p^2\ |\ n$ for some prime $p$. Show that there may exists at most half of elements $a$ in the multiplicative group of integers modulo $n$, such that ...
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1answer
26 views

Q: $\alpha: H \rightarrow \operatorname{Aut}(H)$ a nontrivial homomorphism. Must $H \rtimes_{\alpha} H \ncong H \times H$?

$H$ is a group and $\alpha: H \rightarrow \operatorname{Aut}(H)$ is a nontrivial homomorphism. Does $H \rtimes_{\alpha} H \ncong H \times H$ necessarily? This is a follow-up to this thread. ...
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14 views

The Variety X(K) as a subset of the analytification of X

I am working with Sam Payne's artical Analytification is the limit of all tropicalizations, and because of my limited understanding of analytification it gives me some difficulties. The article: ...
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27 views

Explicit matrix representation of an algebraic extension

This may be considered an extension of this question. Let $\mathbb{F}$ be a field, and let $p(X)\in\mathbb{F}[X]$ be an irreducible polynomial. Let $\mathbb{F}_p$ be the extension of $\mathbb{F}$ by ...
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35 views

Showing that $1-a$ has a multiplicative inverse in a ring $R$ [duplicate]

Question Let a belong to a ring R with unity and suppose that $a^{n}=0$ for some positive integer n. Prove that $1-a$ has a multiplicative inverse in $R$. Hint: $$\left ( 1-a \right ...
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1answer
48 views

Extension of DVRs and uniformizers

Let $(A,\mathfrak m)$ be a regular, Noetherian, local, domain of dimension $2$ and consider a prime ideal $\mathfrak p\subset A$ of height $1$. Moreover let $\hat{A}$ be the completion of $A$ with ...
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2answers
42 views

Power of a polynomial in Galois field

Let $f(x) \in GF[q](X)$, where $q = p^m$ and $p$ prime. Is the following true? $$f^{p^m}(X) = f(X^{p^m}).$$ I tried to prove the assertion above and got stuck at the following: $$ \begin{align} ...
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1answer
41 views

Maximal ideals in a ring of sets

A ring $R$ is called Boolean if $x^2 = x$ for all $x \in R$. It follows that Boolean rings have characteristic $2$ and are commutative. Let $S$ be a non-empty set, then $P(S)$ with $A + B = (A - B) ...
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1answer
38 views

$n$th root of power series when its coefficients are from a field with positive characteristic

Let $k$ be algebraically closed field of characteristic $p>0$. Let's consider a power series $f(x,y)\in k[[x,y]]$. Under what conditions (on $n$, $f$, ...) there exists $g(x,y)\in k[[x,y]]$ such ...
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1answer
23 views

How to find normal subgroups from a character table?

I know that normal subgroups are the union of some conjugacy classes Conjugacy classes are represented by the the columns in a matrix How could we use character values in the table to determine ...
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50 views

How many normal groups does $\mathbb{Z}_n \times \mathbb{Z}_m$ have?

I am currently studying abstract algebra, and I found this problem. I can determine the number of elements and the maximum order of the same. I need to find the subgroup H to apply the theory, but how ...
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1answer
41 views

$p$-groups have normal subgroups of each order [duplicate]

Suppose $|G|=p^n$. Then $G$ has a normal subgroup of order $p^m$ for every $0\le m\le n$. By induction. It is clearly true for $n=0$. Now suppose $k<n$ and $H_i$ is a normal subgroup of $G$ of ...
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37 views

Trivial intersection of quotient subgroups

Suppose that $H/P$ and $M/P$ are two subgroups of a group $G/P$ such that intersect trivially i.e. $(H/P) \cap (M/P) = \{1_{G/P}\}$ = $\{P\}$. Is it true that $H \cap M = P?$
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Valuation fields and linear disjointness

I've got a question about linear disjointness of extension fields. Here I refer to the definition that Wikipedia gives: Two algebras $A$, $B$ over a field $k$ inside some field extension $\Omega$ ...
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Algebraic K-theory: induced maps

Let $f:A \to B$ be a homomorphism of rings. One way to define algebraic $K$ theory ($K_0$ group) is to consider the Grothendieck group of isomorphism classes of all finitely generated projective ...
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1answer
44 views

Representation of elements in a field

Let $A$ and $B$ be integrals domains, such that $A$ is integral over $B$. Writing $K(A)$ for the field of fractions, suppose that $K(A)$ is generated over $K(B)$ by a single element in $A$, say ...
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1answer
25 views

Dimension of a direct sum of characters (example with $S_3$)

Here is the character table of $S_3$: I was wondering how one can determine the dimension of for example the sign character $sgn$. Could we get it from the character table? Also, if we define $A$ ...
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1answer
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Example of an ideal which is not principal in the ring $\mathbb{Z} [x]$ [duplicate]

Give an example of an ideal in the ring $\mathbb{Z} [x]$ is not principal. What kind of example would be the easiest?
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1answer
23 views

Why if $Z/m\oplus Z/n = Z/mn$ then $(n,m)=1$ [duplicate]

Prove that if $Z/m\oplus Z/n = Z/mn$ then $(n,m)=1$. I have proved the converse, but here there is something I am missing. Hints instead of full answers are appreciated. Thanks.
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If $H$ and $K$ are nilpotent normal subgroups then $C(HK)$ is non trivial

I know that this follows from the fact that $HK$ is nilpotent but maybe there is an easier way to proof this? I wanted to show that there is an Element in $HK$ that commutes with $H$ and $K$. I ...
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The rigid additive tensor category freely generated by an object

I've found myself to be absolutely mystified by something in Deligne and Milne's notes on Tannakian categories. Namely, on p. 16 they are showing that there is a rigid additive tensor category ...
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46 views

Forgetful functor applied to a module

I try to find a left adjoint to the forgetful functor $U: R-Mod \longrightarrow Ab$. I considered a functor $F:Ab \longrightarrow R-Mod$ defined by $F(G)=Hom(U(R),G)$. I'm not so sure that in this ...
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2answers
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Extension of intersection of ideals

Let $f:A \rightarrow B$ be ring homomorphism and $\mathfrak{a}_1,\mathfrak{a_2}$ be ideals of $A$. Let $\mathfrak{a}^e$ denote the extension of an ideal $\mathfrak{a}$ of $A$ in $B$. An exercise shows ...
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1answer
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Elements in a Field of size $27$

I constructed the Field $$F_3[x]/<1 + 2x + x^3>$$ as the question asked to construct a field of size $27$ and I understood everything up to this point. The solution then says the elements in ...
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1answer
33 views

Solving a linear system in $\mathbb{Z}_{12}$?

Let $\alpha \in \mathbb{Z}_{12}$. I need to solve the following system in $\mathbb{Z}_{12, +, \cdot}$ for every $\alpha$ : \begin{cases} 6x + 5y = 0 \\ 8x + y = \alpha \end{cases} I'm confused because ...
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The simplest reference for Wedderburn decomposition

Assume $k$ is a field and $A$ is a $k$-algebra then the Wedderburn decomposition says $A=A_{sep}\oplus Nil(A)$, $A_{sep}\rightarrow A_{red}$ via $a\mapsto \bar{a}:=a+Nil(A)$. Where $A_{sep}$ means ...
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Arithmetic with congruence classes

I need to compute the following expression in $\mathbb{Z}_{5, +, \cdot}$ : $$ [2]_5^4 - [4]_5^4 \cdot [3]_5^4 \cdot [2]_5^4 $$ I'm not sure what is the best way to do this. Should I determine all the ...
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Polynomial Path [duplicate]

Let $x(t)$ and $y(t)$ be real polynomials in $t$. Show that there is always a polynomial relation $f(x,y)=0$. This question is taken from Artin, Algebra, Chapter 3 Vector Spaces. I have no idea how ...
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1answer
41 views

$GL_2(\Bbb C)$ acts on a certain set

Let $G:=GL_2(\Bbb C)$, $B$ and $T$ be the subgroup consisting of all upper triangular and diagonal matrices in $G$, respectively. Set $w:= \left( \begin{array}{cc} 0 & 1\\ 1 & 0 ...
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What does it mean for a representation to be one-dimensional?

For example, take the Heisenberg Lie Algebra with the following basis: $X=\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$ $Y=\begin{bmatrix} 0 ...
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On the question of the Galois group of some polynomial. [duplicate]

I want to ask you some question on the Galois group of some polynomial. Let $p_1<p_2<\cdots<p_n$ be distinct prime numbers. Let's consider $f(t)=(t^2-p_1)(t^2-p_2)\cdots(t^2-p_n)\in ...
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1answer
19 views

Lie algebra homomorphism and representation

I am solving a multiple part problem on Lie algebra representations. I have done the first three parts, but am stuck on part (iv) as follows: Define a linear map $\phi : \mathbb{g} \rightarrow ...
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1answer
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Determining the number of subgroups of $\Bbb Z_{14} \oplus \Bbb Z_{6}$

I want to determine how many subgroups does the additive group $G:=\Bbb Z_{14} \oplus \Bbb Z_{6}$ have? There are many related posts in our site, for instances: here and there. However, it ...
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Existence of algebraically closed extensions

Just want to check everything is fine. Basically, the point is to first take some polynomial ring with some huge amount of variables so that each can become a suitable root when we project it in the ...
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Minimal presentation of a tensor product of modules

Let $(R,m)$ be a local commutative noetherian ring. Suppose I have two modules $M$ and $N$ over $R$ given in terms of minimal presentations $$ M = \operatorname{coker}A, $$ and $$ N = ...
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2answers
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Are the statements about the free $R$-module correct? [closed]

Let $R$ be a commutative ring with unit. If $F$ is a free $R$-module with finite rank, does it hold that each set of its generators contains a basis and that each linearly independent set of ...
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48 views

Show that they are isomorphic

Let $R$ be a commutative ring with unit and $M$ be a $R$-module. I want to show that $\text{Hom}_R(R,M)\cong M$ as $R$-modules and $\text{Hom}_R(R,R)\cong R$ as rings. $$$$ I have done the ...