Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, and other advanced topics.

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15
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1answer
80 views

Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z[x]$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$?

Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z[x]$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$? I think there are no such polynomials, but how to prove?
0
votes
0answers
21 views

A question about separable extension

We say $K$ is separably generated over $k$ if there exists a transcendence basis $\{x_i,i\in I\}$ of $K/k$ such that $K/k(x_i,i\in I)$ is a separable algebraic extension. We say $K$ is separable over ...
5
votes
2answers
39 views

Prove that $k(\alpha+\beta)=k(\alpha,\beta)$

I am trying to solve the following problem: Let $k$ be a finite field and let $k(\alpha,\beta)/k$ be finite. If $k(\alpha)\cap k(\beta)=k$, prove that $k(\alpha,\beta)=k(\alpha+\beta)$. What I ...
0
votes
1answer
37 views

Intuitive explanation for a map $z \to z^p$.

Let $p$ be prime and let $G$ be the group of $p$-power roots of 1 in $\mathbb{C}$ . Prove the map $z \to z^p$ is a surjective homomorphism. $\textbf{My Attempt:}$ $G=\{ z \in \mathbb{C} \mid ...
1
vote
2answers
28 views

Composition of ring homomorphism

I have three rings $A,B,C$ and ring homomorphisms $f: A \rightarrow B$ and $g: B \rightarrow C$, which are both surjective. Is it true that $C$ is isomorphic to $$ A / (\ker(f), \ker(g)) ? $$ If so ...
3
votes
1answer
56 views

How to prove this innocent looking isomorphism

I've got a Dedekind domain $R$ with quotient field $K$, a non-zero prime ideal $P$ of $R$. I form the completion $\widehat{K}$ of $K$ wrt the valuation $v_P$ associated to $P$. Let $\widehat{R}$ be ...
9
votes
4answers
134 views

Does every group act faithfully on some group?

Cayley's Theorem shows that every group acts faithfully on some set. In other words, one can find an injective group homomorphism $\sigma: G\to S_{A}$ where $S$ is the set of all bijections on some ...
0
votes
0answers
16 views

Finite-dimensional algebras with invertible elements and without idempotents

Does there exist a finite-dimensional algebra containing a nontrivial invertible element, with no nontrivial idempotents? By 'nontrivial', I mean not proportional to the identity.
4
votes
2answers
94 views

If a group has no maximal subgroups then all elements are non-generators? Frattini subgroup characterization

This question is the last leg of an exercise I've been working on in which we characterize the intersection of all maximal subgroups as the subgroup of all non-generators. I've already shown that if a ...
1
vote
2answers
32 views

Jacobson radical in $A[x]$ where $A$ is a ring.

Let $A$ be a ring, and $A[x]$ be the ring of polynomials in an indeterminate x, with coefficients in A. I found several proofs online that in this case the Jacobson radical equals to the nilradical, ...
3
votes
1answer
28 views

calculating signature and showing group homomorphism

I got this question in my group theory book. I think I understand the theory behind it but can't seem to use it to get a solution im happy with. Let $V = M_2(\mathbb F)$. For $x,y \in V$ define ...
5
votes
3answers
41 views

Let $\mathbb Q\le K$ finite extension of fields $\alpha\in K$. Then $N(\alpha)=\pm1\Leftrightarrow\alpha$ is a unit

Let $\mathbb Q\le K$ (where $K\le\mathbb C$) be a finite extension (say $|K:\mathbb Q|=n$) of fields and let $\alpha\in K$ be an algebraic integer, i.e. $\alpha$ is a root of a monic polynomial over ...
2
votes
1answer
36 views

Show that $(x_1\times\dots\times x_k)\times x_{k+1}=x_1\times\dots\times x_{k+1}$

Let $a\times b\times c$ denote $a\times(b\times c)$. Given $(a\times b)\times c=a\times(b\times c)$, how do you prove $$(x_1\times\dots\times x_k)\times x_{k+1}=x_1\times\dots\times x_{k+1}$$ ? ...
-2
votes
0answers
34 views

Order of the elements of a right coset [on hold]

What can we say about order of the elements of a right coset in a finite group.
3
votes
1answer
45 views

Linear representation of a free group

I need to prove the following: "Prove that the free group of rank 2 is linear." So to my best understanding, and please correct me if I'm wrong: I actually need to show a homomorphism from the free ...
7
votes
1answer
119 views

A question about Sylow subgroups

Let $G$ be a finite group and $P\neq\{e\}$ be a Sylow $p$-subgroup of $G$ and $P^g\neq P$ be its conjugate in $G$. If we know that $P\cap P^g\neq \{e\}$, can we conclude that $Z(P)\cap Z(P^g)\neq ...
2
votes
2answers
48 views

Ring Structure: Definition

Let $R = \{a+bi\mid a,b \in \mathbb Z, i^2=-1\}$, with addition and multiplication defined by $(a+bi)+(c+di)=(a+c)+(b+d)i$ and $(a+bi)(c+di)=(ac-bd)+(bc+ad)i$, respectively. (a) Verify that $R$ is ...
0
votes
0answers
33 views

Finite generated abelian group $G$ and $H<G$. What is the rank of $(G/H)/(G/H)_t$?

I saw another question about this problem here. However there are quite different answers from my expectation. Anyway, here are my trials. Trial 1 : By structure theorem, $G\cong G_t\oplus F_1$ ...
0
votes
0answers
24 views

About product order [on hold]

Are there any references talking about product order on this wikipeida link? thanks!
0
votes
1answer
35 views

Linear algebraic group

Let A be a finite dimensional algebra over C . This means that there is a multiplication map $f : A \times A \to A$ that is bilinear ( it is not assumed to be associative). Define the automorphism ...
1
vote
1answer
21 views

Proving that finite direct and inverse limits exist in an additive category having kernels and cokernels

I have attempted to prove this but am unable to complete the proof. Below is my attempt. Let $\mathcal{A}$ be a category satisfying the conditions in the title and $\{M_i,\phi^i_j\}$ be a finite ...
5
votes
0answers
78 views

Units of $\mathbb{Z}[\sqrt[4]2]$

How would one compute the units in $\mathbb{Z}[\sqrt[4]2]$? According to one source, it can be shown that the fundamental units are $1 + \sqrt[4]2$ and $1 + \sqrt{2}$, but it does not specify the ...
0
votes
1answer
60 views

Can I assign a distinct homomorphism to every function?

I consider an algebraic structure and particular structure preserving morphisms (think groups and group homomorphism). I wonder if I can assign a distinct such morphisms to every function between any ...
0
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0answers
42 views

can somebody provide me this paper by NIVAN? [on hold]

equations in quaternions by I. Nivan published in AMM Vol 48 , 654-661 (1941). please i need it, and could not find it online anywhere.
0
votes
2answers
23 views

Conjugates and normalizer

Let $H=\langle(1 2 3)\rangle$ and let $G=S_3$. Now, $$(1 2)(1 2 3)(1 2)=(1 3 2)=(1 2 3)^{-1}$$ Since $(1 2)$ conjugates a generator of $H$ to another generator of $H$, we can conclude that $(1 2) ...
2
votes
2answers
38 views

Show that $A = \{(3x,y)~|~ x,y \in Z\}$ is a maximal ideal of $Z \oplus Z$. My Attempt Shown

Show that $A = \{(3x,y)~|~ x,y \in Z\}$ is a maximal ideal of $Z \oplus Z$. Here's my attempt, Please tell me where did I go wrong. Attempt: When $R$ is a commutative ring with unity and $I$ is any ...
0
votes
0answers
30 views

$p$-completion of a $\mathbb{Z}_p$-module

Let $p$ be a prime number, $\mathbb Z_p$ the ring of $p$-adic integers. Let $M$ be a finitely generated $\mathbb Z_p$-module and $\widehat{M}$ its $p$-completion $\varprojlim_n M/{p^n M}$. ...
0
votes
0answers
28 views

Must the index $k=|G:HC_G(x)|$ be finite?

I want to solve the following Exercise from Dummit & Foote's Abstract Algebra text: Assume $H$ is a normal subgroup of $G$, $\mathcal{K}$ is a conjugacy class of $G$ contained in $H$ and $x ...
2
votes
1answer
31 views

Action of GL$(2,\mathbb{R})$ on symmetric matrices

This is a problem from an old qualifier. Let GL$(2,\mathbb{R})$ act on SYM, the real symmetric 2x2 matrices, via $S \mapsto A^T SA$ for $A \in$ GL$(2,\mathbb{R})$ and $S \in$SYM. Show that each ...
1
vote
0answers
23 views

$p$-completion of a $\mathbb{Z}$-module

Let $p$ be a prime number, $\mathbb Z_p$ the ring of $p$-adic integers. Let $M$ be a finitely generated $\mathbb Z$-module and $\widehat{M}$ its $p$-completion $\varprojlim_n M/{p^n M}$ which is a ...
2
votes
2answers
71 views

Localization does not commute canonically with infinite direct products

Let $S=\mathbb{Z}-\{0\}$, and the fraction ring \begin{equation} S^{-1}\prod_{1}^{\infty}\mathbb{Z}_{i}=\{\frac{(a_{1},a_{2},...,a_{n},...)}{b}:b,a_{i}\in\mathbb{Z},b\neq 0\}.\end{equation} Show ...
0
votes
1answer
23 views

Let $R[X]$ be a polynomial ring over the field $R$. Suppose $q(X) = w(X) \in R[X]$. How can I show $q(a) = w(a)$ for $a \in R$?

Let $R[X]$ be a polynomial ring over the field $R$. Suppose $q(X) = w(X) \in R[X]$. How can I show $q(a) = w(a)$ for $a \in R$ ? To give a specific example. Suppose $q(X) = a_nX^n+...+ ...
4
votes
1answer
41 views

Projective modules over $kG$ equivalent to injective.

Let $k$ be a field and $G$ is finite group. I want to prove that a $kG$ module $P$ is projective iff it's injective. I proved that if module is projective then it's injective. 1) $kG$ is injective ...
1
vote
1answer
40 views

Proof that $~\hom_G(V,\bigoplus U)=\bigoplus\hom_G(V,U)$

Yesterday I have already asked for help in this post to prove that $~\hom_G(V,\bigoplus U)=\bigoplus\hom_G(V,U)$ for finite $\bigoplus$. Unfortunately, I did not fully understand the answer because it ...
5
votes
1answer
37 views

If $0$ is the zero-object $ \Longrightarrow F(0) $ is the zero object when $F$ additive

Let $$ F : \text{A-Mod} \to \text{A-mod} $$ be an additive functor. Then if $0$ is the zero-object $F(0) $ is the zero object. Why this is true ? The definition of additive functor that I know is ...
4
votes
2answers
39 views

Suppose that $R$ is a commutative ring and $|R|=30$. If $I$ is an ideal of $R$ and $|I|=10$, prove that $I$ is maximal ideal

Suppose that $R$ is a commutative ring and $|R|=30$. If $I$ is an ideal of $R$ and $|I|=10$, prove that $I$ is maximal ideal Solution: $|R/I|=3 \implies R/I \approx Z_3$ which is a field. If $R$ is ...
2
votes
0answers
35 views

How many (unordered) bases does $\Bbb F_q^n$ have as a vector space over $\Bbb F_q$?

This is a practice problem from a Tier 1 exam, and I want to check that my reasoning is correct. We shall first consider how many different ordered bases $\Bbb F_q^n$ has. Recall that $|GL_n(\Bbb ...
3
votes
1answer
44 views

what is this property called?(Field theory)

In what references the following property $P$ of a field $F$ is investigated? The property $P$: For all $n\in \mathbb{N}$ if $\sum_{i=1}^{n} f_{i}^{2}=0,\;\;f_{i}\in F$ then $f_{i}=0, \forall i\in ...
5
votes
1answer
55 views

$L\otimes_{\Delta}\text{Hom}_{\Delta}(M,\Delta)\cong \text{Hom}_{\Delta}(M,L)$

This is exercise 5 in maximal orders by I.Reiner. This is not homework though. Let $\Delta$ be a ring $L_{\Delta}$ be any module, and let $M_{\Delta}$ be a finitely generated and projective. ...
1
vote
1answer
54 views

What is the injective envelope for $\mathbb{Z}/n\mathbb{Z}$.

In the category of $\mathbb{Z}-$modules, what is the injective envelope of $\mathbb{Z}/n\mathbb{Z}$. I was hoping to find a divisible group containing $\mathbb{Z}/n\mathbb{Z}$ such that it is also ...
3
votes
1answer
33 views

If an integral domain $R$ has a factorization basis, is it a UFD?

By a factorization basis for an integral domain $R$, let us mean a subset $\xi$ of the commutative monoid $R^\times = R \setminus \{0\}$ such that firstly, no two elements of $\xi$ are associates, and ...
2
votes
1answer
50 views

groups generated by two elements of order 3

I'd like to know whether it is possible to find a characterization (cardinality?) of the set of finite, non-abelian groups generated by two elements $a$ and $b$ whose order is $3$? Is it the same task ...
1
vote
2answers
45 views

n-torsion elements of $\mathbb{Q}/\mathbb{Z}$

How does one show that the elements of $\mathbb{Q}/\mathbb{Z}$ annhilated by $n$ can be represented by $\lbrace \ [0], [1/n],[2/n], \ldots , [(n-1)/n] \ \rbrace \subset \mathbb{Q}/\mathbb{Z}$
0
votes
2answers
31 views

order of dihedral

I am learning abstract algebra, and I don't quite understand the order of the symmetry of dihedral. When you look at a squares, I agree that there will be 8 symmetry. But all the operations have cycle ...
-3
votes
2answers
30 views

If $F_3 =\mathbb{ Z}/3\mathbb{Z}$, show that $F_3$ is a field. How can this be done? [on hold]

If $F_3 = \mathbb{Z}/3\mathbb{Z}$, show that $F_3$ is a field. How can this be done? Please help! $\mathbb{Z}=\{ \text{set of integers}\}$.
3
votes
1answer
26 views

Examples where there is no power integral basis

Let $A$ be a completion discrete valuation ring with quotient field $K$, $L/K$ finite and separable, and $B$ the integral closure of $A$ in $L$. Let $P, \mathfrak P$ be the unique maximal ideals of ...
0
votes
2answers
43 views

Show that if K is a non-zero ideal of Z/mZ,

Show that if K is a non-zero ideal of Z/mZ, then K is the principal idea. Please help!
0
votes
1answer
29 views

Constructing an Algebraically Closed Space

How would one construct the simplest -- or canonical, if a canonical construction exists -- countably infinite algebraic closure of the natural numbers?
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0answers
29 views

About 2nd isomorphism theorem or diamond isomorphism theorem

Given group G=MN where M and N are normal subgroup of G. I can show that $\frac{G}{M\cap N}$ isomorphic to $\frac{G}{M}\times\frac{G}{N}$. However I am a bit confused when I draw the lattice. I got ...
0
votes
1answer
46 views

Possible Fields?

Is there an algebraically closed field which is a 1-dimensional vector space (as opposed to complex numbers which are 2-D)? Also is there a complete $\aleph_0$ field?