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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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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1answer
15 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
26 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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17 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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40 views

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

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

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

Does $ (-1)^2 = 1$ anywhere where you have associativity and an inverse element? Thanks!
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1answer
28 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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45 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
29 views

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
20 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
39 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
39 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 $k-1$ ...
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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
22 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
35 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
23 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
36 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
27 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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33 views

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
64 views

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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37 views

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
58 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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31 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
21 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 ...
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1answer
46 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 ...
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1answer
66 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 ...
2
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1answer
59 views

$\operatorname{Hom}_R(\mathfrak{a},M)$ is isomorphic to $\mathfrak{a}^{-1}M$ if $R$ is a Dedekind domain

I want to prove Lemma 2.5.1 of Silverman's Advanced Topics in The Arithmetic of Elliptic Curves (whose proof is left to the reader): Let $R$ be a Dedekind domain, let $\mathfrak{a}$ be a ...
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1answer
29 views

Power series ring $k[[x]]$ contains elements transcendental over $k(x)$

If $k$ is countable, then $k[x]$ is countable and it seems easy to figure out $k[[x]]$ has elements transcendental over $k(x)$ because $k[[x]]$ is uncountable by using the fact that algebraic closure ...
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0answers
56 views

Example of $A$-module but not $A$-algebra. [duplicate]

If $A$, $B$ are commutative rings, and if $B$ is an $A$-algebra then it is also an $A$-module. I am looking for an example that shows that the converse is not true. That is, I am looking for ...
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44 views

Homomorphisms from $\mathbb{C}$ to $M_2(\mathbb{R})$

Let $\phi_1$ and $\phi_2$ be two injective ring homomorphisms from $\mathbb{C}$ to $M_2(\mathbb{R})$. Show that there exists a $g\in GL_2(\mathbb{R})$ such that $\phi_2(x) = g\phi_1(x)g^{-1}$ for all ...
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1answer
43 views

What is the difference between submodules of $A/\mathfrak a$ as an $A$-module or as an $A/\mathfrak a$-module?

If $\mathfrak a$ is an ideal of unital commutative ring $A$, Then we can see to $A/\mathfrak a$ as an $A$ module or as an $A/\mathfrak a$ module. If $A=\mathbb Z$ there is no structure difference ...
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1answer
46 views

Show that every short exact sequence $0\rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ (with $M''$ free) is split exact

Note: $M,M',M''$ are modules. In order to show that it's split exact, I understand that I need to show that $\beta:M\rightarrow M''$ has a left inverse, and similarly that $\alpha:M'\rightarrow M$ ...
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recover (pontrjagin) ring structure from the localization (w.r.t. $\pi_0$)

Let $R$ be a ring and $S$ a given multiplicative subset of $R$. Suppose we know the multiplication structure of $S$. If we know the ring structure of $R[S^{-1}]$, the localization of $R$ with respect ...
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21 views

Permutation calculator

I am studying the Mathieu group $M_{12}$ on the twelve letters $\infty,7,6,8,X,2,0,3,4,1,9,5$ (in this specific order) in the form that it is generated by the permutations $(0123456789X)$, ...
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1answer
55 views

Proof that $\sqrt{3} \notin \mathbb{Q}(\theta)$ where $\theta^4-2=0$. [on hold]

This is a problem in Robert Ash's lecture notes in Algebraic Number Theory. I have to prove that $\sqrt{3} \notin K=\mathbb{Q}(\theta)$ where $\theta^4-2=0$, using the fact that ...
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39 views

Can we always make a group? [duplicate]

Can we always find an operation on non-empty set, which create a group$?$ I cann't imagine, how not, but is it proof for that?
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6Sz as the automorphism group of the complex Leech lattice

Consider the Leech lattice as a complex lattice over the Eisenstein integers. Both in Conway ("Sphere packings, lattices and groups") and Wilson ("The Complex Leech Lattice and Maximal Subgroups of ...
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1answer
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For a group epimorphism $f : G \to H$ with kernel $K$, prove that $G \simeq K \rtimes H$. Why is $G \simeq K\times H$ if $G$ is abelian? [on hold]

For a group epimorphism $f : G \to H$ with kernel $K$, prove that $G \simeq K \rtimes H$. Why is $G \simeq K\times H$ if $G$ is abelian? This question is from group theory in Abstract Algebra and ...
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3answers
446 views

Example of a ring with infinitely many zero divisors and finitely many invertible elements

I am preparing to my abstract algebra exam and I try to find an example of a ring with infinitely many zero divisors and finitely many invertible elements (rather simple if possible). Does it even ...
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5answers
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Why is the commutator defined differently for groups and rings?

The commutator of two elements in a group is defined as $[g, h] = g^{−1}h^{−1}gh.$ In a ring, the commutator of two elements is $[a, b] = ab - ba.$ I'm asking because a ring is a (abelian) group ...
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2answers
32 views

Three polynomials as unknowns of an equation

If three polynomials $f,g,h\in\mathbb R[x]$ are such that $[f(x)]^2 –x[g(x)]^2+[h(x)]^2=0$, what can we conclude about $f, g, h$?
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1answer
18 views

classification of groups of order $4p, p\ge 5$, need help finding automorphism

So I've been working on this problem for my qual prep class, and I have it all down except for one detail. I'm doing it by semidirect products, and with the Sylow $p$ group normal, choosing the ...
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0answers
33 views

Projective general linear group on discrete valuation ring

Let $R$ be a complete discrete valuation ring and $k$ its residue field. Let $H$ be a finite subgroup of $PGL_2(k)$ such that its order is prime with char($k$). Is there some elementary way to show ...
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34 views

Covering groups

I am studying the Steiner system $S(5,6,12)$ and the ternary extended Golay code $\mathscr{C}_{12}$. The automorphism group of the Steiner system is the Mathieu group on twelve elements $M_{12}$ ...
3
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1answer
106 views

What motivates the definition of a ring in abstract algebra? [on hold]

I've read a lot of questions about rings but I still don't know why they are useful/ I'm sure they are, but I don't know why. Are their properties somehow used in proofs or as foundations for ...
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0answers
49 views

Valuations on $\Bbb Q(t)$

Ex. $2.3.3$ in Algebraic Number Theory by Neukirch is the following: Let $k$ be a field and $K = k(t)$ the function field in one variable. Show that the valuations $v_{\mathfrak p} $ associated to ...
2
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2answers
33 views

There is no nontrivial ring homomorphism between two commutative rings with unity and characteristic of distinct primes

The following is an old exam question and the question is: Show that there is no nontrivial ring homomorphism between two commutative rings with identity if their characteristics are distinct primes. ...