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Questions tagged [abstract-algebra]

Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, and modules, among other topics. Whenever you use this tag, please also include related tags like group-theory, ring-theory, modules, etc., in order to clarify which topic of abstract algebra is most related to your question and also to help other users find similar questions through the search engine.

36
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683 views

If polynomials are almost surjective over a field, is the field algebraically closed?

Let $K$ be a field. Say that polynomials are almost surjective over $K$ if for any nonconstant polynomial $f(x)\in K[x]$, the image of the map $f:K\to K$ contains all but finitely many points of $K$. ...
31
votes
0answers
411 views

What is $\tau(A_n)$?

Suppose G is a finite group. Define $\tau(G)$ as the minimal number, such that $\forall X \subset G$ if $|X| > \tau(G)$, then $XXX = \langle X \rangle$. What is $\tau(A_n)$? Similar problems for ...
26
votes
0answers
891 views

Where Fermat's last theorem fails

It's fairly well known that Fermat's last theorem fails in $\mathbb{Z}/p\mathbb{Z}$. Schur discovered this while he was trying to prove the conjecture on $\mathbb{N}$, and the proof is an application ...
21
votes
0answers
413 views

Non-reflexive module isomorphic to its double dual

Could you give me an example of a non-reflexive module isomorphic to its double dual? I found an example here but I cannot understand it, do you have any simpler examples? By this question we should ...
19
votes
0answers
175 views

Why $E\otimes_KE\cong EG$ implies that Galois theory works?

This is a part of statement in the book I do not fully appreciate. Suppose $E/K$ is Galois extension and $G$ the galois group of $E/K$. $E[G]$ is the group ring formed by finite group $G$. "It is ...
19
votes
0answers
303 views

Galois correspondence and characteristic subgroups

It is well-known that Galois correspondence sends a normal subgroup to a normal extension of a field. Specifically, given a Galois extension $L/K$ and the corresponding Galois group $G$, normal ...
19
votes
0answers
815 views

Show that in any group of order $23 \cdot 24$, the $23$-Sylow subgroup is normal.

Show that in any group of order $23 \cdot 24$, the $23$-Sylow subgroup is normal. Let $P_k$ denote the $k$-Sylow subgroup and let $n_3$ denote the number of conjugates of $P_k$. $n_2 \equiv 1 \mod ...
19
votes
0answers
416 views

How much do idempotent ultrafilters generate in terms of semigroups?

It is known that the set of ultrafilters on, say, the natural numbers $\mathbb{N}$, can naturally be endowed with the structure of a compact topological left semigroup (which fails to be anything ...
17
votes
0answers
145 views

Can you make the triviality of $\langle a,b,c \mid aba^{-1} = b^2, bcb^{-1} = c^2, cac^{-1} = a^2 \rangle$ more trivial?

I recently learned the following pleasant fact. (It was in the proof of Proposition 3.1 of this paper - but don't worry, there's no model theory in this question.) Let $G$ be a group, and let $a,b,c\...
16
votes
0answers
221 views

Can we find a bound so that we can conclude $G$ is a $p$-group?

Let $n_p$ be number of the elements of order $p$ in a group $G$. My motivation is that if $n_2\ge\dfrac 34 |G|$ then $G$ is $2$-group. You can check it from this. Is there such general bound for $...
15
votes
0answers
478 views

Is there a field $F$ which is isomorphic to $F(X,Y)$ but not to $F(X)$?

Is there a field $F$ such that $F \cong F(X,Y)$ as fields, but $F \not \cong F(X)$ as fields? I know only an example of a field $F$ such that $F$ isomorphic to $F(x,y)$ : this is something like $F=k(...
14
votes
0answers
280 views

If $K\cong K(X)$ then must $K$ be a field of rational functions in infinitely many variables?

If $k$ is any field, then the field $K=k(X_0,X_1,\dots)$ of rational functions in infinitely many variables satisfies $K(X)\cong K$ (by mapping $X$ to $X_0$ and $X_n$ to $X_{n+1}$). My question is, ...
14
votes
0answers
209 views

**UNSOLVED** Find an integer $\geqslant2$ that is build up out of only $1$'s and $0$'s in base $1,\;\ldots,\;10$.

This riddle bothers me for a few weeks now and I'm starting to worry that I need some $p$-adic Number theory to solve this. I solve most of the riddles in a day, but this one is just annoying to me. I ...
14
votes
0answers
429 views

Galois Group of Composite Field vs. Second Isomorphism Theorem

$\DeclareMathOperator{\Gal}{Gal}$ In my abstract algebra class, we learned about how Galois groups interact with composite fields. Namely, if $K/F$ is Galois, and $L/F$ is any extension: $$\Gal(KL/L) \...
13
votes
0answers
222 views

UFD containg a special element

Does anyone know an example of a unique factorization domain $R$ that is (i) not a Dedekind domain (or equivalently, not a principal ideal domain) and (ii) contains some irreducible element $r \in ...
13
votes
0answers
363 views

Groups of order $180$, $540$, $1080$ are not simple.

Here's how I solve the problems. Thanks for pointing out what might be the weakness of my solutions. Actually, what I want are other ways of solving this kind of problems, appart from counting the ...
13
votes
0answers
233 views

The possible degrees of $\mathbb{Q}(a,b)$ in terms of the degrees of $a$ and $b$

Let $n,m \geq 1$ be natural numbers. Is there a characterization of those natural numbers $d$ for which there are algebraic numbers $a,b$ of degrees $n,m$ such that $\mathbb{Q}(a,b)$ has degree $d$ ...
13
votes
0answers
621 views

Character theory of $2$-Frobenius groups.

Edit Summary: I've posted this on MO and received a partial answer there. Can anybody help me expand on this? Definition. Let $G$ be a finite group and $F_1=\text{Fit}\,G$ and $F_2/F=\text{Fit}\...
12
votes
0answers
1k views

Induced representation, Ind(Res(U))

I am reading a book of Fulton and Harris "Representation theory, a first course". Now it's all about representation theory of finite groups, and there is one exercise, which I can't solve: If $U$ is ...
12
votes
0answers
221 views

Free medial magmas

A medial magma is a set $M$ with a binary operation $*$ satisfying $(a*b)*(c*d) = (a*c)*(b*d)$. Medial magmas constitute an algebraic category $\mathsf{Med}$, therefore there is a functor $\mathsf{Set}...
11
votes
0answers
226 views

Are all verbal automorphisms inner power automorphisms?

Suppose $G$ is a group. $\DeclareMathOperator{\Wa}{Wa}\DeclareMathOperator{\Tame}{Tame}\DeclareMathOperator{\Aut}{Aut}$ Lets call $\phi \in \Aut(G)$ verbal automorphism iff $\exists n \in \mathbb{N}, ...
11
votes
0answers
403 views

What is the most general algebraic structure that a finite set has?

For an object $X$ in a category with finite products, define its endomorphism Lawvere theory to be the Lawvere theory generated by $X$: its $n$-ary operations are given by $\text{Hom}(X^n, X)$, and so ...
11
votes
0answers
153 views

$\text{SL}(2, \mathbb{F}_q)$, for which characters is the $G$-representation irreducible?

Followup to here. Let $\mathbb{F}$ be a finite field with $q$ elements, and let $G = \text{SL}_2(\mathbb{F})$. The group $G$ acts linearly on the $2$-dimensional vector space $\mathbb{F}^2$ and fixes ...
11
votes
0answers
336 views

What properties do the rings of infinite, upper-triangular matrices have?

I'm very familiar with the ring of $n\times n$ matrices over a field, the ring of $n\times n$ upper triangular matrices over a field, and the ring of infinite column-finite matrices over a field. But ...
11
votes
0answers
1k views

The multiplicative group of all complex $2^n$-th roots of unity, where $n = 0 , 1, 2 , \ldots$

Let $G$ be the multiplicative group of all complex $2^n$-th roots of unity, where $n = 0 , 1, 2 , \ldots$. Then assess the following claims: Every proper subgroup of $G$ is finite. $G$ has a ...
11
votes
0answers
88 views

Let $F_\infty=\bigcup_{n\geq1}\operatorname{Q}(2^{1/2^n})$ , $K_\infty=\bigcup_{n\geq1}\operatorname{Q}(\zeta_{2^n})$, what is the intersection?

Let $F_\infty=\bigcup_{n\geq1}\operatorname{Q}(2^{1/2^n})$ and $K_\infty=\bigcup_{n\geq1}\operatorname{Q}(\zeta_{2^n})$. What is the intersection $F_\infty\cap K_\infty$? (Here $\zeta_{2^n}$ is a ...
11
votes
0answers
231 views

When does a polynomial fixing a subring imply its coefficients are in that subring?

Let $S$ be a subring of $R$. If $p$ is a polynomial with coefficients in $S$, then $p$ fixes $S$ (as a function, that is, $p(s)\in S$ for all $s\in S$). A converse statement is: If $p$ is a ...
11
votes
0answers
572 views

cohomological proof of Maschke's theorem

I have been working on the following problem.. I have spent plenty of time trying to solve it myself. I am, however, unable to prove one small step in the argument. Beneath you can find my attempt. ...
10
votes
0answers
2k views

Was Atiyah's proof of the odd order (Feit-Thompson) theorem false?

I read last year that Atiyah thought he had found a proof of the odd order theorem of only 12 pages, using $K$-theory, and that people were trying to figure out if it was correct or not. But I never ...
10
votes
0answers
162 views

How to effectively compute fundamental units in rings?

Consider the ring of integers of $K=\mathbb{Q}(\sqrt 2,\sqrt 3)$. By Dirichlet's unit theorem the units of $\mathcal O_K$ have rank 3, so they are expressible as $\pm u_1^au_2^bu_3^c$ for suitable ...
10
votes
0answers
147 views

Is there an elementary way to prove that the algebraic integers are a Bézout domain?

Well, the title of my question says all, but let me give some context. Right now I'm writing some lecture notes on ring theory with a little of commutative algebra. I wrote a few results about ...
10
votes
0answers
148 views

Polynomials with same image on the rationals?

I am struggling with this problem: If $P_1$ and $P_2$ are two polynomials such that $P_1(\mathbb{Q})=P_2(\mathbb{Q})$, show that $P_1(x)=P_2(ax+b)$ for some constants $a,b$. Here is what I have done....
10
votes
0answers
942 views

What is “Field with One Element”?

I was reading the Wikipedia article about The Field with One Element and I came across the following quotes: "...F1 refers to the idea that there should be a way to replace sets and operations, the ...
10
votes
0answers
516 views

Converse of Chinese Remainder Theorem

Chinese Remainder Theorem for commutative rings with identity Let $R$ be a commutative ring with identity. If $I, J$ are ideals of $R$ satisfying $I+J=R$, then there is an isomorphism of rings: $$R/(...
10
votes
0answers
177 views

Whether a functor is exact?

I am stuck with exercise $1$ of section $3$ of chapter $1$ in the book Cohomology of number fields by Neukirch. The exercise is to show that the functor from $A \rightarrow C^n(G,A)$ is exact, where $...
10
votes
0answers
438 views

algebraically closed fields of characteristic 0 and $\mathbb{C}$

Let $k$ be an algebraically closed field of characteristic 0. Then I've heard that if $k$ has cardinality no greater than that of $\mathbb{C}$, then there is an embedding $k\hookrightarrow\mathbb{C}$....
10
votes
0answers
390 views

Proving a ring in which $r^n=r$ for all $r$ is commutative.

Let $R$ be a ring with identity such that there is a positive integer $n\geq 2$ for which $r^n=r$ for all $r\in R$. Prove $R$ is commutative. I had proven before that If $n=2$ it is commutative as ...
10
votes
0answers
571 views

On a Proof in Galois Theory

We have the following lemma (used in the proof of Abel's Theorem): If $\text{Char}(F)=0$, $E/F$ is a radical extension, and $K/E$ is the Galois closure of $E/F$, then $K/F$ is also a radical ...
9
votes
0answers
132 views

Rings with 'non-harmless' zero-divisors

The following excerpt is from pp. 246–247 of Paolo Aluffi's Algebra: Chapter 0: 1.2. Prime and irreducible elements. Let $R$ be a (commutative) ring [with $1$], and let $a,b\in R$. We say that $a$ ...
9
votes
0answers
85 views

Finding a chain map from an $F$-acyclic resolution to an injective resolution which is a monomorphism in each degree

Let $\mathcal{A}$ be an abelian category with enough injectives. Let $F:\mathcal{A}\to\mathcal{B}$ be a left-exact additive functor to $\mathcal{B}$ another abelian category If $M$ has an $F$-...
9
votes
0answers
227 views

A Ramanujan-type trigonometric identity

At the end of the following article: http://www.ijpam.eu/contents/2013-85-1/15/15.pdf It is asserted that the russian mathematician, Sergey Markelov, in private communication, told them that he ...
9
votes
0answers
154 views

Every polynomial's image contains $0$ or $1$ in a field $\Bbb F$

This question talks about fields in which every polynomials are almost surjective, while I am interested in the following case: $\Bbb F$ is a field such that for every non-constant polynomial $f$ ...
9
votes
0answers
151 views

Subgroups of $GL_n$ and group actions

For my abstract algebra class I have to do some exercises concerning group actions of $GL_n(K)$ on the set of flags $$F_n = \{0 \subseteq V_1 \subseteq V_2 \subseteq \ldots \subseteq V_r = K^n \,\vert\...
9
votes
0answers
117 views

UCT and Künneth, Hom-Tensor adjunction

A few days ago, there was a similar question in an other context. Künneth: Consider the tensor product of modules $H_i(X;\mathbb{Z})\otimes A$, where $A$ is an abelian group, $X$ is a topological ...
9
votes
0answers
103 views

Affine group, $[L : \mathbb{Q}] = n\varphi(n)$ or ${1\over2}n\varphi(n)$

Let $L$ be the Galois closure of $K = \mathbb{Q}(\sqrt[n]{a})$, where $a \in \mathbb{Q}$, $a > 0$ and suppose $[K : \mathbb{Q}] = n$. How do I see that $[L: \mathbb{Q}] = n\varphi(n)$ or ${1\over2}...
9
votes
0answers
364 views

Prove that the group $\mathrm{GL}(n, \mathbb{Z})$ is finitely generated

Knowing that for $n \geq 2$, $\mathrm{GL}(n, \mathbb{Z}) = \big\{ A \in \mathrm{M}_{n,n}(\mathbb{Z}) \mid \det(A) \in \{ 1, −1 \} \big\}$ is a group with respect to matrix multiplication, prove that ...
9
votes
0answers
151 views

In which algebraic theories do 'free' and 'projective' coincide?

Free models of algebraic theories are always projective objects in the category of models, but the converse is not always true. For instance, some (actually, all) projective modules are direct ...
9
votes
0answers
148 views

Classification of all subrings

Let $R$ be an integral domain whose underlying additive group is finitely generated free and whose field of fractions $K$ is a finite Galois extension of $\mathbb{Q}$. Is there a method of ...
9
votes
0answers
52 views

Objects whose limiting behaviour resembles a group

Is there a name for a structure that isn't a group, but that begins to behave like a group the more operations are performed? I'm trying to take the idea of an attractor from dynamical systems and ...
9
votes
0answers
784 views

A More Advanced Version Of Aluffi

Paulo Aluffi's Book, Algebra, Chapter 0 aims to teach basic algebra from a categorical viewpoint. The first chapters of the book, however, introduce groups and rings using very basic categorical ...