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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Is $ f_n=\frac{(x+1)^n-(x^n+1)}{x}$ irreducible over $\mathbf{Z}$ for arbitrary $n$?

In this document on page $3$ I found an interesting polynomial: $$f_n=\frac{(x+1)^n-(x^n+1)}{x}$$ Question is whether this polynomial is irreducible over $\mathbf{Q}$ for arbitrary $n \geq 1$ ? In ...
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295 views

What is Spec of the Adeles?

Let $K$ be a global field and $A_K$ the ring of adeles. What are the prime ideals of $A_K$? I have been told that a full proof of this is quite subtle, but have been unable to find a reference ...
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181 views

Can any commutative ring of characteristic $p\in\mathbb P$ be written as the form $R/(p)$ with $R$ being a ring of characteristic $0$?

Let $S$ be a commutative ring with identity with $\operatorname{char}S=p$, where $p$ is a prime number. I wonder if we can always find a ring $R$ such that $\operatorname{char}R=0$ and $R/(p)\cong ...
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224 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 ...
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100 views

How smooth can non-nice associative operations on the reals be?

Suppose ${*}:\mathbb R\times\mathbb R\to\mathbb R$ is $\mathcal C^k$ and associative. Does it necessarily satisfy the identity $a * b * c * d = a * c * b * d$? For $k=0$ the answer is "no" -- a ...
13
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365 views

Values attained by $|G/Z(G)|$?

So I was working through some problems in a book on $p$-groups and noticed that $p$-groups have some really nice properties. So I started computing what the values of $|G/Z(G)|$ for $p$-groups. I ...
13
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220 views

What's the latest on Laver tables?

A couple of years ago, I was astonished and delighted to learn about Laver tables, a sequence (indexed on $n$) of Cayley-like tables for a binary operation $\star$ on numbers $i,j\leq 2^n$ that ...
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613 views

The radical solution of a solvable 17th degree equation

(The question is at the bottom of the post.) Here's a "natural" solvable 17-th deg eqn with small coefficients: $$\begin{align*} x^{17}-6 x^{16}&-24 x^{15}-42 x^{14}-31 x^{13}-23 x^{12}-7 ...
12
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122 views

A “generalized field” with $q$ elements, when $q$ is any number?

It is well-known that if a finite field has $q \in \mathbb{N}$ elements, then $q$ is prime power and $q > 1$. However, various modification of the concept of a "field" have been made in order to ...
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68 views

Analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$.

We consider an analogue of the Dirichlet $L$-function in $\mathbb{F}_q[T]$. Let $g \in \mathbb{F}_q[T]$, $g \neq 0$, let $\chi: (\mathbb{F}_q[T]/(g))^\times \to \mathbb{C}^\times$ be a homomorphism, ...
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192 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
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158 views

Cogroup structures on the profinite completion of the integers

Let $\mathsf{ProFinGrp}$ be the category of profinite groups (with continuous homomorphisms). This is equivalent to the Pro-category of $\mathsf{FinGrp}$. Notice that $\widehat{\mathbb{Z}} = ...
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316 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 ...
11
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432 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 ...
10
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107 views

Intuitive/geometric way of thinking about effective divisors?

What is the motivation/intuition/geometric way of thinking about an effective divisor? I know that a divisor is effective if all its coefficients are non-negative. We write $D \ge 0$ for ...
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91 views

Study of rings of the form $R+I$

In my life I saw lots of ways of constructing rings: polynomial rings, quotient rings, localizations, endomorphism rings, rings of fractions, integral closure of a ring, center of a ring, etc... These ...
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111 views

Proving that the number of integer solutions of $x^2-Ny^2=1$ is infinite

I am trying to prove that the number of integer solutions of $x^2-Ny^2=1$ is infinite whenever N is a squarefree integer. For this I define norm of $a+b\sqrt N=a^2-Nb^2$. Now I prove that $a+b \sqrt ...
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101 views

Multiplicity of point as a zero of polynomial.

Let $C \subset \mathbb{P}_2$ be a projective curve defined by a homogeneous polynomial $P(x, y, z)$, and let $L \subset \mathbb{P}_2$ be the line $\{z = 0\}$. Assume $L$ is not a component of $C$. If ...
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88 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 ...
10
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260 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 ...
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310 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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135 views

Free medial magmas

A medial magma is a set $M$ with a binary relation $*$ satisfying $(a*b)*(c*d) = (a*c)*(b*d)$. Medial magmas constitute an algebraic category $\mathsf{Med}$, therefore there is a functor $\mathsf{Set} ...
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230 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 ...
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89 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 ...
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143 views

A group acting on functions of functions of functions

Given a group acting on a set $X$, there is a standard way to define an action of the group on the set of functions of $X$. This can be extended to the set of functions of functions of $X$ as I show ...
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71 views

What can we learn about a magma by studying these monoids?

Given a magma $(X,*)$, we get three monoids in the following way. First, define a pair of functions $L,R : X \rightarrow (X \rightarrow X).$ $$(Lx)(y) = x*y,\quad (Rx)(y) = y*x$$ Then each of the ...
9
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307 views

$G$ is finite, $A \leq G$ and all double cosets $AxA$ have the same cardinality, show that $A \triangleleft G$

If $G$ is a finite group and $A$ is a subgroup of $G$ such that all double cosets $AxA$ have the same number of elements, show that $gAg^{-1}=A$ for all $g \in G$. Here is my attempt, I guess it's ...
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251 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 ...
9
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366 views

Trivial intersection of algebraic sets?

The question came up while reading a bit more into the Hilbert-Zariski theorem I asked about the other week. Suppose $V$ is an algebraic variety over arbitrary field $k$. (For this situation, I'll ...
9
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328 views

On group automorphism of subgroup a group $G$

Let $G$ be a group and $H$ be a subgroup of $G$. When is $\rm{Aut}(H)$ a subgroup of $\rm{Aut}(G)$?
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131 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 ...
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Genus of $k(T)$?

Let $F$ be a function field in one variable with total constant field $k$, let $X$ be the set of all places $F$, and let $S$ be a nonempty finite subset of $X$. Then the genus of $F$ is equal to the ...
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123 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 ...
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277 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 ...
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136 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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218 views

Morita-invariance of Hochschild (co)homology.

Ok, I'm reading this paper by Christian Kassel on associative algebras and Hochschild (co)homology and on page 19 he says that Hochschild homology is Morita-invariant, by which he means that if $R$ ...
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250 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 ...
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37 views

$|G:H|=p^n$ means $O_p(H)\leq O_p(G)$?

Let $H\leq G$ (finite group) and $|G:H|=p^n$, ($p$ is a prime number) prove that: $$O_p(H)\leq O_p(G)$$ note: $O_p(G)$ defined as the intersection of all Sylow-$p$ groups in $G$ I try to prove ...
7
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29 views

Number of elements of the group $SL_6(\mathbb{Z}/p^k\mathbb{Z})$

For $p$ a prime, what is the number of elements of the group $SL_6(\mathbb{Z}/p^k\mathbb{Z})$, $k \ge 1$? I can answer the $k=1$ case. For each element of $SL_6(\mathbb{Z}/p\mathbb{Z})$, there are ...
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141 views

Eigenvalues of a quasi-circulant matrix

The following matrix cropped up in a model I am building of a dynamical system: $$A= \begin{bmatrix} 1 - \alpha & \alpha/2 & 0 & 0 &\cdots & 0 & 0 & \alpha/2\\ \alpha/2 ...
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index 2 subgroups of the infinite product of Z/2Z

Is it possible to describe all the index 2 subgroups of the group $G = \prod_{i\in \mathbb{N}}\; \mathbb{Z}/2\mathbb{Z}$? For example, one can take the kernel of the $i$-th projection map ...
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78 views

Is it the case that each $\Phi^q$ of a stable cohomology operation $\{\Phi^q\}$ is a natural homomorphism?

So as the question statement asks, is it necessarily the case that each $\Phi^q$ of a stable cohomology operation $\{\Phi^q\}$ is a natural homomorphism? I suspect the answer is yes, but I don't know ...
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83 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 ...
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117 views

Classify groups of order 171

This is a problem from Stanford Algebra Qualifying Exam, Fall 1998. I know the standard way is to use Sylow theorems and semidirect product. $171 = 9\cdot 19$. By Sylow theorems, $n_3|19$ and ...
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66 views

The difference between the ring version and module version of Chinese Remainder Thereom.

Chinese Remainder Theorem for Commutative Rings If $R$ is a commutative ring with $1$ and $I, J$ are ideals of $R$ that are pairwise coprime or comaximal (meaning $I + J = R$), then $IJ = I \cap J$, ...
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188 views

Why differential Galois theory is not widely used?

E.R.Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation. My ...
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34 views

The order of subgroup generated by two distinct element?

$G$ is a group of order $2$6, if $x$ and $y$ are two distinct element of order $2$, what could the order of $\langle x,y\rangle$ be? By lagranges theorem, $\langle x\rangle$ and $\langle y\rangle$ ...
7
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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 ...
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174 views

Why the Steinberg idempotent is idempotent?

Consider the group $GL_n(\mathbb{F}_p)$. We have the following subgroups : -$\Sigma_n$ the symmetric group (permutation matrices) -$B_n$ the Borel subgroup (upper triangular matrices) -$U_n$ the ...
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255 views

Is there some sort of correspondence between groups and partitions of a set?

Every group action on a set $S$ partitions the set into orbits. Conversely, for every partition of $S$ is there a group action such that the set of orbits of the group action equals the partition? ...