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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Infinitude of the primes $p=1 \operatorname{mod} n$

$\textbf{Theorem:}$ Fix $1 < n \in \mathbb{Z}$. There are infinitely many primes $p=1 \operatorname{mod} n$. $\textbf{Proof}$ Recall that the $n$-th cyclotomic polynomial $\Phi_n(x)$ is monic, has ...
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29 views

What is the Galois group of $\mathbb{Q}_p[\zeta] / \mathbb{Q}_p$, where $\zeta$ is a $p^r$th root of unity?

Let $\zeta$ be a $p^r$-th root of unity and $\mathbb{Q}_p$ the p-adic numbers. I would like to know the Galois automorphisms of $\mathbb{Q}_p[\zeta] / \mathbb{Q}_p$. I already know, the degree of ...
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23 views

Ring Structure for Non Commutative Groups: Is there a grander reason for Abelian requirements?

So I was considering the following idea. Let a generalized Ring $gR$ be a set equipped with two operations $$u_1, u_2$$ Such that $gR$ is a with the operation $u_1$ and the operation $u_2$ ...
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1answer
9 views

dimension of the quotient of a bialgebra

I am stuck in a proof of a lemma that I am in need of. The situation is as follows: Let $k$ be a field and $A$ and $B$ two finite-dimensional $k$-bialgebras, where the dimension of $A$ is a prime ...
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11 views

How to choose a left-add$(X)$-approximation with a certain property

Let $A$ be an artin algebra and $X,Y$ in mod $A$. Suppose, $0\rightarrow Y \stackrel{\alpha}{\rightarrow} X^n\stackrel{\beta}{\rightarrow} X^m$ is exact. Set $C:=Coker(\alpha)$ (as module) and ...
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35 views

When is $M\otimes N$ a module?

So suppose we have a $R$-$S$-bimodule $M$, and an $S$-$T$ bimodule $N$, then we can construct the abelian group $M\otimes_S N$. Under what conditions could we make this a module? I would expect this ...
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1answer
14 views

Every modular right ideal is contained in a modular maximal ideal

If $R$ is a ring, possibly without $1$, a right ideal $\mathfrak{a}$ of $R$ is modular if there exists $e\in R$ such that $r-er\in \mathfrak{a}$ for all $r\in R$. So $e$ is a left identity mod ...
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342 views

What's a group whose group of automorphisms is non-abelian?

I recently attended an interview for admission to graduate programs in Mathematics. The interviewing professor asked me a question - Tell me a group whose group of automorphisms is non-abelian. ...
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13 views

Finding binary operations on connected graphs

If $G = (V,E)$ is a connected graph with $||V|| \geq 2$ , $W(G)$ being the set of all paths in $G$. How do you find a binary operation $ +$ on $W(G)$ such that $\langle W(G),+\rangle$ is an algebra ...
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2answers
45 views

Proving associativity in Algebra

How to proof that a specially defined Transitive Join for the relations $R \subseteq A$ x $B$ und $S \subseteq B$ x $C$ is associative? The join is defined as: $R \Join S =_{def} \{(a,c)| $ there is ...
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21 views

Maximum order of an element in Alternating group of degree 10.

What is the maximum order of any element in $A_{10}$? My attempt: I tried this problem. But I am not sure about the answer. My answer is 21 because 10 can be written as 7+3. $A_{10}$ can have ...
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1answer
15 views

Ring of polynomials as free module

Is it true that $R=k[x,y]$ is a free $R$-module ? I think that it isn't true. Natural candidate for the base is $\{x^{\alpha}y^{\beta}\}_{\alpha,\beta}$, but : $x\cdot (xy) +(-y)\cdot x^2 =0$ and ...
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1answer
19 views

Find the integral closure of an integral domain in its field of fractions [duplicate]

Let $k$ be a field and let $R = k[x,y]/(x^2-y^2+y^3)$. Note that $R$ is an integral domain. Let $F$ be the field of fractions of $R$. How to determine the integral closure of $R$ in $F$? I have ...
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10 views

Find how many elements [on hold]

Find how many elements in a group of order 30 has the order 5,and explain the reasons. Cant do it. Any ideas?
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1answer
45 views

Finding a binary operation on $\{1, \ldots, n\}$ so that each $k$ has exactly $k - 1$ left inverses

What is an example of a binary operation on a set $\{1, \dots n\}$ so that each element $k \in \{1 \dots n\}$ has respectively $k-1$ left inverse elements? I have been trying various combinations ...
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1answer
14 views

Question in proof from James Milne's Algebraic Number Theory

I'm having difficulty understanding a step in a proof from J.S. Milne's Algebraic Number Theory (link). Here $\zeta$ is a $p$th root of unity and $\mathfrak p = (1-\zeta^i)$ for any $1\leq i\leq p-1$ ...
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93 views

What exactly is Hensel doing for us in this result?

I'm reading a paper where the author appeals to Hensel's lemma, but it is not clear to me quite how it is meant to be applied (or, for that matter, which version!). My commutative algebra background ...
2
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1answer
36 views

Example of subgroup of $\mathbb Q$ which is not finitely generated

I was looking for the proper subgroup of $\mathbb Q$ which is not finitely generated under the addition operation. We know every finitely generated subgroups of $\mathbb Q$ is finitely cyclic. For ...
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2answers
64 views

Find all of the homomorphisms $ \varphi: \mathbb{Z}_{15} \to \mathbb{Z}_{6} $.

I'd like to find all of the homomorphisms $ \varphi: \mathbb{Z}_{15} \to \mathbb{Z}_{6} $. What I've tried so far: I know that $ |\text{Im} (\varphi)| $ divides $ ...
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1answer
45 views

Basic question on the free algebra

Let $k$ be a field and consider the (unital and associative) free algebra on $k$ with two generators ($x$ and $y$), $A= k < x,y >$. I have two basic questions concerning this algebra: 1) If ...
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1answer
26 views

Find all the possible pairs of submodules $M_1$ and $M_2$ of the $\mathbb{Z}$-module $\mathbb{Z}_{18}$ so that $\mathbb{Z}_{18} = M_1 \oplus M_2$

I started by considering the possible proper subgroups of $\mathbb{Z}_{18}$, which are $\langle\bar{9}\rangle$, $\langle\bar6\rangle$, $\langle\bar3\rangle$ and $\langle\bar2\rangle$, which are also ...
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What is the formalism for a map that returns the adjacent vertex positions of a given adjacency matrix?

How do I formally denote a map that returns the adjacent vertex positions of a given adjacency matrix? Example: V = undirected adjacency matrix $$ V = \left[ \begin{matrix} v_{1,1} & v_{1,2} ...
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17 views

Image of $x$ under canonical projection is root of polynomial.

Let $M(x)$ be an irreducible polynomial in $K[x]$ where $K$ is a field. Let $I$ be the ideal generated in $K[x]$ by $M(x)$. Let $\alpha$ be the image of $x$ in the field $J= K[x] / \langle M(x) ...
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1answer
31 views

Why is the $\text{End}_A(M)$-module $\text{Hom}_A(N,M)$ finitely generated?

Let $A$ be an Artin algebra and let $M,N$ be some finitely generated modules in mod(A). Why is then the $\text{End}_A(M)$-module $\text{Hom}_A(N,M)$ finitely generated? Thanks for the help.
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23 views

extend trace and norm from a number field $K$ to $ K \otimes_{\mathbb Q} \mathbb R $

I read somewhere that we can extend the trace and the norm of a number field $K$ to the commutative algebra $ V=K \otimes_{\mathbb Q} \mathbb R$. Before state exactly my question, let me write the ...
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3answers
54 views

Prove that $G=\{z \in \mathbb{C}: |z|=1\}$ is an abelian group [on hold]

Prove that $G=\{z \in \mathbb{C}: |z|=1\}$ is an Abelian group with the multiplication operation of complex numbers.
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1answer
36 views

converging power series over $p$-adic integers is a UFD

Denote by $|\cdot |$ the $p$-adic norm, and let $$\mathbb Z_p \{z\}=\left\{\sum_{n=0}^\infty a_nz^n;\ a_n\in \mathbb Z_p ,\ |a_n|\underset {n\rightarrow \infty} {\longrightarrow 0} \right\}$$ the ...
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121 views

In what structures does $ (-1)^2 = 1$?

Does $ (-1)^2 = 1$ anywhere you have associativity and an inverse element? Thanks!
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39 views

Cyclotomic polynomials, properties.

Let $F$ be a field of characteristic prime to $n$, and let $F^a$ be an algebraic closure of $F$. Let $\zeta$ be a primitive $n$th root of unity in $F^a$. I know that the monic polynomial $\Phi_n(X)$ ...
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2answers
54 views

Let $G,*$ a group and $a,b,c,d \in G$. Prove that …

Let $G,*$ a group and $a,b,c \in G$. Prove that the equation $x*a*x*b=x*c$ it has a unique solution in $G$. Ideas? I do not know where to start. D =
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2answers
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Issue with associativity of group

Given $G=(1,2)\subset R$ and the operation $x∗y = \frac{3xy-4x-4y+6}{2xy-3x-3y+5}$ Prove that $(G,∗)$ is an abelian group. So here's my issue with this. For it to be a group I must prove that: ...
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1answer
28 views

Sets of compositions of homomorphisms

I am looking of a relation in the form: $$ Hom(X,Z) = Hom(X,Y)\otimes Hom(Y,Z), $$ or: $$ Hom(X,Z) \subseteq Hom(X,Y)\otimes Hom(Y,Z), $$ or similar (maybe it's not a tensor product? maybe the ...
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1answer
46 views

Galois extension and prime number.

Let $G$ be a finite group with order $n$, i.e., $|G|=n$. Show that there is a prime number $p\geq n$ and a finite Galois extension $L/K$ with $Gal(L/K)\approx G$ and $[K:\mathbb{Q}]=p!/n$. Honestly, ...
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1answer
52 views

Binary operations in an algebra

Is there a binary operation ° for the algebra <{1,...,n},°> such that for each $k \in \{1,...,n\}$ there are exactly $k-1$ ...
3
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1answer
39 views

Looking for a terminology in ring theory (“ideal” which is not necessarily closed under addition )

I am wondering if there is a name for the subsets $S$ of a commutative ring $R$ such that for every $r\in R$ and every $s\in S$ we have $rs\in S$. Thus $S$ is ...
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1answer
26 views

Relations between $R^fG$ and either $\mathbb{C}^fG$ or $\mathbb{Z}^fG$.

Denote by $RG$ the group ring of the group $G$ over the commutative ring $R$. A result by Passman saying that if $R$ is a commutative ring then $$RG=R\otimes_{\mathbb{Z}}\mathbb{Z}G.$$ As a result, ...
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2answers
36 views

Characterizing the A-module M/S

I've been working through this for a little while, and I'm not 100% sure I understand what I'm supposed to be doing here, or maybe I'm not grasping correctly what they mean by "Characterize". ...
2
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1answer
28 views

Matrix $B \in M_n(S)$, for $S$ an $R$-algebra, with $R$-independent entries, $A \in GL_n(R)$. Are the entries of $AB$ $R$-independent?

Let $R$ be a field (or a domain, or a commutative ring), and $S$ an $R$-algebra. Let $B \in M_n(S)$ have $R$-independent entries. Let $A \in GL_n(R)$. Are the entries of $AB$ $R$-independent? I ...
2
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1answer
41 views

Analysing Exact Sequence

I have the following exact sequence $\mathbb{Z}\xrightarrow{f}\mathbb{Z} \xrightarrow{g} K_0(\mathcal{T})\xrightarrow{h}\mathbb{Z}\xrightarrow{0}0$. From here I want to conclude that ...
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1answer
31 views

Ring homomorphism of tensor product of algebras

Let $B, C$ be two $A$-algebras, $f:A \to B, g: A\to C$ the corresponding ring homomorphisms. From this we can construct an $A$-algebra $B \otimes _A C$ and the mapping $ a \mapsto f(a) \otimes ...
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1answer
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Polynomial-closed properties of rings

If $R$ is a ring with certain property, sometimes when we pass to the polynomial ring in one variable, the ring $R[x]$ still has the same property. For instance, it's a theorem that if $R$ is a UFD ...
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3answers
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Is there a unique homomorphism of $\mathbf Z $ into $A$?

I am reading Atiyah's Itroduction to Commutative Algebra. On pages 30, he say that ii) Let $A$ be any ring, Since $A$ has an identity element there is a unique homomorphism of the ring of ...
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Some properties of finite subgroup G of multiplicative group $F^*$ [on hold]

F is a field. $\psi_G(d)$ is number of elements with order d. N(F)- set of all zeros of polynomial $X^d-1 \in F$ and $|G|=m$. For $d\in \mathbb{N}$ and $\psi_G(d)\neq 0$ Show that: $d|m$ and ...
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1answer
24 views

Verify that $A \oplus B$, where $A$ and $B$ are cyclic groups of orders 2 and 3, is the cyclic group of order 6

Let's define $A$ and $B$ as follows: $A$ = {e,a} $B$ = {e,b,2b} Then $A\oplus B= \{\{e+e\},\{e+b\},\{e+2b\},\{a+e\},\{a+b\},\{a+2b\}\}$ which is equal to ...
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1answer
60 views

Aut$(G)\cong \Bbb{Z}_8$

I am looking for a group such that Aut$(G)\cong \Bbb{Z}_8$. Obviously Aut$(\Bbb{Z}_n)\ncong \Bbb{Z}_8$ for any $n$. Also Aut$(D_4)\cong D_4$, neither symmetric/alternating groups are of any help ...
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0answers
33 views

Set of functions from a finite field to the integers

Has the set of functions $\mathbb{F}_q \to \mathbb{Z}$ endowed with pointwise addition and additive/multiplicative convolution been studied? Does anyone know a reference or keywords to search for? ...
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1answer
19 views

*$G$-invariant* symmetric bilinear form & $G'=\Bbb Z_2\times\Bbb Z_2$.

I got a problem with the last point I solved all the points, from (a) to (h), but I have no idea how to solve (i): how can I associate a bilinear form to a represtation? What is a $G$-invariant ...
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2answers
22 views

Basis of a field extension

Let $K$ be a field, and let $A$ be a $K$-algebra such that $\alpha \in A$. Then the natural homomorphism $$ \phi: K[x] \to K[\alpha], \hspace{3mm} (x \mapsto \alpha )$$ has a kernel which is a ...
5
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1answer
47 views

General notions of basis

Free groups, free abelian groups, and vector spaces all have a notion of 'basis': a subset $B$ of the structure such that everything in the structure can be written uniquely as a finite combination of ...
4
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1answer
67 views

Under what conditions does $M \oplus A \cong M \oplus B$ imply $A \cong B$?

This question is fairly general (I'm actually interested in a more specific setting, which I'll mention later), and I've found similar questions/answers on here but they don't seem to answer the ...