Questions tagged [abstract-algebra]

For questions about monoids, groups, rings, modules, fields, vector spaces, algebras over fields, various types of lattices, and other such algebraic objects. Associate with related tags like [group-theory], [ring-theory], [modules], etc as necessary. To clarify which topic of abstract algebra is most related to your question and help other users when searching.

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103 votes
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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 $K$. ...
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102 votes
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Ring structure on the Galois group of a finite field

Let $F$ be a finite field. There is an isomorphism of topological groups $(\mathrm{Gal}(\overline{F}/F),\circ) \cong (\widehat{\mathbb{Z}},+)$. It follows that the Galois group carries the structure ...
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33 votes
0 answers
931 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\...
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30 votes
1 answer
642 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 ...
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25 votes
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796 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, ...
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24 votes
0 answers
608 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 ...
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23 votes
0 answers
293 views

If $G \bigoplus H$ is isomorphic to a proper subgroup of itself, then must the same be true of one of $G$ and $H$?

Let $G$ and $H$ are groups. If $G \bigoplus H$ is isomorphic to a proper subgroup of itself, then must the same be true of one of $G$ and $H$? (at least H or G) I found some examples of $G$ such ...
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22 votes
0 answers
846 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) \...
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19 votes
1 answer
285 views

Existence of rational sequence such that a polynomial is split over $\Bbb{Q}$

Does there exist a sequence $(a_n)_{n\in \Bbb{N}}$ of rationals such that for all $n\in \Bbb{N}$, $a_n\neq 0$ and the polynomial $a_0+a_1X+\cdots+a_nX^n$ is split over $\Bbb{Q}$? I was asked this ...
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18 votes
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582 views

Is there a group theoretic proof that $(\mathbf Z/(p))^\times$ is cyclic?

Theorem: The group $(\mathbf Z/(p))^\times$ is cyclic for any prime $p$. Most proofs make use of the fact that for $r\geq 1$, there are at most $r$ solutions to the equation $x^r=1$ in $\mathbf Z/(p)$...
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17 votes
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418 views

Is $z=e^{\frac{1}{\log(x)}}$ a solution to a polynomial?

Is $z=e^{\frac{1}{\log(x)}}$ a solution to a polynomial? $x\in\Bbb Q~\setminus\{0,1\}.$ Wolfram is telling me $z$ is algebraic but I'm not sure that I believe this. I believe it would follow from ...
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17 votes
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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(...
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17 votes
0 answers
385 views

Free medial magmas

A medial magma is a set $M$ with a binary operation $*$ satisfying $$(a*b)*(c*d) = (a*c)*(b*d)$$ for all $a,b,c,d \in M$. Medial magmas constitute a finitary algebraic category $\mathsf{Med}$, ...
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16 votes
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721 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}\...
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16 votes
1 answer
614 views

Is $\bigl(X(X-a)(X-b)\bigr)^{2^n} +1$ an irreducible polynomial over $\mathbb{Q}[X]$?

Let $a, b \in \mathbb{Q}$, with $a\neq b$ and $ab\neq 0$, and $n$ a positive integer. Is the polynomial $\bigl(X(X-a)(X-b)\bigr)^{2^n} +1$ irreducible over $\mathbb{Q}[X]$? I know that $\bigl(X(X-...
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15 votes
0 answers
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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 ...
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15 votes
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746 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 ...
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15 votes
0 answers
285 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....
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15 votes
2 answers
1k views

Advanced Galois theory/field theory book suggestions

I have done the equivalent of an undergraduate algebra course(rings/fields/groups) and have read Stewart's Galois theory book. Galois theory was a lot of fun and I would like to continue studying it ...
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14 votes
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153 views

Understanding $(x)=(y)$ in rings that aren't domains

Suppose $R$ is a commutative ring with $1$. I would like to get a better understanding of the equivalence relation on $R$ $$x\sim y\iff(x)=(y)\iff\exists u,v\in R,~x=uy~\text{ and }~y=vx$$ where $(x)$ ...
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14 votes
0 answers
270 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 ...
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  • 931
14 votes
0 answers
257 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 $d$ ...
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  • 2,354
14 votes
0 answers
176 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 ...
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14 votes
0 answers
768 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/(...
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14 votes
0 answers
2k 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: ...
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14 votes
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350 views

Why is the Quasitriangular Hopf algebra called "Quasitriangular"?

The precise definition of a Quasitriangular Hopf algebra can be found on wikipedia. What is the reason behind the word "Quasitriangular"? Is it because the R-matrix is a triangular matrix, ...
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14 votes
1 answer
2k views

$\mathbb{Q}(\sqrt{23})$ 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 $\mathbb{Z}[\sqrt{...
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  • 141
13 votes
0 answers
176 views

Is $\mathrm{U}(\mathfrak{a} \oplus \mathfrak{b}) \cong \mathrm{U}(\mathfrak{a}) \otimes \mathrm{U}(\mathfrak{b})$ over a commutative ring?

Let $\mathbb{k}$ be a commutative ring and let $\mathfrak{g}$ be a Lie-Algebra over $\mathbb{k}$. Suppose that $\mathfrak{a}$ and $\mathfrak{b}$ are two Lie subalgebras of $\mathfrak{g}$ such that $\...
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13 votes
0 answers
199 views

Generic behavior amongst "polynomialish" models of $\mathsf{Q}$

Now asked at MO. For $\mathcal{R}=(R_i)_{i\in\mathbb{N}}$ a sequence of countable commutative ordered rings such that $R_i$ is an sub-ordered ring of $R_{i+1}$ and $R_0=\mathbb{Z}$, let $$Poly_\...
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13 votes
0 answers
269 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}, ...
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13 votes
0 answers
199 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 ...
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  • 12.7k
13 votes
0 answers
572 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 ...
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  • 142k
13 votes
0 answers
747 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 $k\hookrightarrow\mathbb{C}$....
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13 votes
1 answer
3k views

Rings with noncommutative addition

I was wondering if "rings" with noncommutative addition are studied at all? Of course, if a ring $R$ has a $1$, then for all $a, b\in R$, $a+a+b+b=(1+1)a+(1+1)b=(1+1)(a+b)=(a+b)+(a+b)=a+b+a+b$, from ...
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12 votes
0 answers
121 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 ...
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  • 361
11 votes
0 answers
449 views

Applications of Jordan-Holder theorem in an abelian category

The Jordan-Holder theorem says that any chain of subobjects of a finite lenght object can be refined to a composition series, and that any composition series has the same lenght. This theorem holds ...
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  • 331
11 votes
0 answers
344 views

Studying representations of orthogonal group via symmetric group?

Let $V$ be the vector space of $n\times n$ symmetric matrices with real entries. On $V$ we have the natural action of the orthogonal group $\textrm{O}(n)$ defined by $g.A:=g\cdot A\cdot g^{t}$ for $g\...
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  • 3,157
11 votes
0 answers
257 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 ...
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  • 5,751
11 votes
1 answer
1k views

Applications of the Dedekind-Hasse criterion

It is a fact that an integral domain $R$ is a principal ideal domain if and only if there is a Dedekind-Hasse function $|R|\setminus\{0\}\xrightarrow{\ \ \delta\ \ }\mathbb{N}$ on $R$, i.e. a function ...
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11 votes
0 answers
586 views

An equivalence between group cohomology and sheaf cohomology

I'm recently reading group cohomology from Serre's book local fields, and he uses there the following terminology $H^q(G,A)$ the q-th degree cohomology of G with coefficent in $A$. So i started to ...
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  • 251
11 votes
0 answers
762 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. ...
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  • 1,380
11 votes
1 answer
656 views

Dual of a finite dimensional algebra is a coalgebra (ex. from Sweedler)

Let $(A, M, u)$ be a finite dimensional algebra where $M: A\otimes A \rightarrow A$ denotes multiplication and $u: k \rightarrow A$ denotes unit. I want to prove that $(A^*, \Delta, \varepsilon) $ ...
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  • 892
11 votes
0 answers
561 views

(Mathematical) Applications of Quantum Group

What are some mathematical applications of Quantum Groups? I have tried researching and found that it is used to find solutions to Yang-Baxter equation. Any other applications? Thanks a lot.
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  • 18.2k
10 votes
0 answers
220 views

Cardinal numbers of right factors of a group

Let $A$ and $B$ be subsets of a group $G$. The product $AB$ is called direct (and we denote it by $A \cdot B$, e.g., see this) if the representation of each element $x$ of $AB$ as $x=ab$, $a\in A$, $b\...
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10 votes
0 answers
457 views

Fermat's Last Theorem ($n=3$) using the Eisenstein integers

I'm doing the first part of the following exercise in Miles Reid's Undergraduate Commutative Algebra: Exercise 0.18: Prove the cases $n=3$ and $n=4$ of Fermat's last theorem. I'm assuming I should ...
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10 votes
0 answers
401 views

Finite algebraic structures where all hyperoperations (addition, multiplication, exponentiation, tetration, etc.) are well-defined

Let $\langle R, +, \times, \uparrow, \uparrow\uparrow, \uparrow\uparrow\uparrow, \ldots; 0, 1\rangle$ be an algebraic structure with two constants $0, 1$ and where an infinite sequence of binary ...
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  • 4,656
10 votes
0 answers
164 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 ...
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10 votes
1 answer
176 views

Liars, adjunctions, and functions $f : S \rightarrow UFS$. Does this lead anywhere interesting?

A student of mine was recently given the following question: "At least one of us is lying," said Andrew. "Only one of us is lying," said Bertas. "Squeak, two of us are lying," said the ...
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  • 65.3k
10 votes
0 answers
130 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}...
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10 votes
1 answer
453 views

Reduction modulo p of a linear group over the rational numbers

A paper (http://arxiv.org/pdf/1407.3158v2.pdf) contains the following theorem: Suppose $\mathbb{G}$ is a connected, simply connected, semisimple algebraic group defined over $\mathbb{Q}$, and let $\...
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