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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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Does every ring of integers sit inside a ring of integers that has a power basis?

Given a finite extension of the rationals, $K$, we know that $K=\mathbb{Q}[\alpha]$ by the primitive element theorem, so every $x \in K$ has the form $$x = a_0 + a_1 \alpha + \cdots + a_n \alpha^n,$$ ...
Eins Null's user avatar
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133 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 $K$. ...
Eric Wofsey's user avatar
113 votes
0 answers
3k views

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 ...
Martin Brandenburg's user avatar
79 votes
0 answers
1k views

Does the $32$-inator exist?

Background It is common popular-math knowledge that as we extend the real numbers to complex numbers, quaternions, octonions, sedenions, $32$-nions, etc. using the Cayley-Dickson construction, we lose ...
pregunton's user avatar
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32 votes
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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, ...
Eric Wofsey's user avatar
31 votes
1 answer
761 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 ...
Chris's user avatar
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28 votes
0 answers
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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 $\mathbf Z/(p)$...
Sam's user avatar
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27 votes
0 answers
666 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 ...
Jakub Konieczny's user avatar
24 votes
1 answer
591 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 ...
Joel92's user avatar
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24 votes
0 answers
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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) \...
Henry Swanson's user avatar
23 votes
0 answers
720 views

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

Is $z=e^{\frac{1}{\log(x)}}$ with $x\in\Bbb Q~\cap(0,1)$ a solution to an algebraic equation? Wolfram is telling me $z$ is algebraic but I'm not sure that I believe this. I believe it would follow ...
zeta space's user avatar
20 votes
0 answers
502 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, ...
yoyostein's user avatar
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19 votes
0 answers
847 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(...
Watson's user avatar
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19 votes
1 answer
304 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 ...
user avatar
19 votes
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411 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 $\mathsf{Med}$, ...
Martin Brandenburg's user avatar
17 votes
0 answers
333 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\...
M.H.Hooshmand's user avatar
17 votes
0 answers
3k 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 ...
frafour's user avatar
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17 votes
0 answers
876 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 ...
Qiaochu Yuan's user avatar
17 votes
0 answers
378 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....
Chris Sanders's user avatar
17 votes
0 answers
971 views

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

Let $k$ be an algebraically closed field of characteristic $0$. Then I have heard that if $k$ has cardinality no greater than that of $\mathbb{C}$, then there is an embedding $k\hookrightarrow\mathbb{...
oxeimon's user avatar
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17 votes
0 answers
775 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}\...
Alexander Gruber's user avatar
  • 27.1k
16 votes
0 answers
669 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 ...
rschwieb's user avatar
  • 155k
16 votes
0 answers
3k 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: ...
user197284's user avatar
16 votes
1 answer
708 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-...
user84673's user avatar
  • 2,027
15 votes
0 answers
275 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 $d$ ...
HeinrichD's user avatar
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15 votes
0 answers
242 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 ...
Arrow's user avatar
  • 13.8k
15 votes
0 answers
886 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/(...
Sungjin Kim's user avatar
  • 20.1k
14 votes
0 answers
178 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 $(x)$ ...
Olivier Bégassat's user avatar
14 votes
0 answers
277 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 ...
Algebrus's user avatar
  • 916
14 votes
1 answer
4k 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 ...
Nishant's user avatar
  • 9,195
13 votes
0 answers
915 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 ...
cansomeonehelpmeout's user avatar
13 votes
0 answers
233 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 $\...
Jendrik Stelzner's user avatar
13 votes
0 answers
203 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_\...
Noah Schweber's user avatar
13 votes
0 answers
279 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}, ...
Chain Markov's user avatar
  • 15.6k
13 votes
0 answers
820 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 ...
Mimmo's user avatar
  • 279
13 votes
0 answers
129 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 ...
HGF's user avatar
  • 371
12 votes
0 answers
278 views

A mistake in Bourbaki's construction of general tensor product?

Fortunately someone seems to have a asked a similar question before here: Bourbaki's construction of generalized tensor product of modules ; to save space and time, I will use the same notation ...
user831160's user avatar
12 votes
0 answers
270 views

when are graded injective modules graded and injective?

Define a graded injective module over a graded ring $R$ to be an injective object in $GrMod-R$ (the category of right graded $R$-modules). From the little research I have done, a graded injective ...
RumDiary's user avatar
  • 121
12 votes
0 answers
365 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 ...
Xam's user avatar
  • 6,129
12 votes
0 answers
384 views

Diophantine equation: $13^x+3=y^2$

$x,y$ are positive integers. $$13^x+3=y^2\iff \left(4+\sqrt{3}\right)^x\left(4-\sqrt{3}\right)^x=\left(y+\sqrt3\right)\left(y-\sqrt3\right)$$ $\gcd\left(y+\sqrt3, y-\sqrt3\right)=1$, therefore $y+\...
user263326's user avatar
11 votes
0 answers
623 views

A big list - Applications of the structure theorem of finitely generated modules over PIDs

I'm now a TA on an undergraduate course "Algebra II" and the main topics of the course are "rings and modules" and "fields and the Galois theory". We shall cover the ...
Hetong Xu's user avatar
  • 2,117
11 votes
0 answers
574 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 ...
less's user avatar
  • 351
11 votes
0 answers
418 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\...
Hans's user avatar
  • 3,571
11 votes
0 answers
548 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 ...
pregunton's user avatar
  • 5,821
11 votes
0 answers
143 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}...
user avatar
11 votes
0 answers
968 views

Description of $\mathrm{Ext}^1(R/I,R/J)$

Let $R$ be a commutative ring with unit and $I$ and $J$ are nonzero ideals of $R$. Do we have a nice description for $\mathrm{Ext}^1_R(R/I,R/J)$? What do I mean by a nice description? For example $$\...
messi's user avatar
  • 1,297
11 votes
0 answers
890 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. ...
yannickvda's user avatar
  • 1,430
11 votes
1 answer
743 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) $ ...
grozhd's user avatar
  • 962
11 votes
0 answers
656 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.
yoyostein's user avatar
  • 19.7k
10 votes
0 answers
214 views

Why do complex analysis and complex geometry exhibit a sort of "arithmetic" behavior?

That is going to be a rather vague question, but as I stated in the title, why do complex analysis and complex geometry seem to exhibit a sort of "arithmetic" behavior, contrary to real ...
user720386's user avatar

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