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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Rotman's exercise 2.8 “$S_n$ cannot be imbedded in $A_{n+1}$”

This question is about the (in)famous Rotman's exercise 2.8 in "An Introduction to the Theory of Groups." I've searched and found similar questions here and in MO, but none of them contains a valid ...
20
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752 views

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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257 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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331 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 ...
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194 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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280 views

On solvable quintics and septics

Here is a nice sufficient (but not necessary) condition on whether a quintic is solvable in radicals or not. Given, $x^5+10cx^3+10dx^2+5ex+f = 0\tag{1}$ If there is an ordering of its roots such ...
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555 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 ...
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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, ...
11
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227 views

Minimal and characteristic polynomials on tensor product spaces

Given two finite-dimensional vector spaces $V$ and $W$ over a common field $k$ as well as $k$-linear transformations $\varphi \colon V \to V$ and $\psi \colon W \to W$, what can be said in general ...
11
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182 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 ...
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150 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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189 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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303 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 ...
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399 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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73 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 ...
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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 ...
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241 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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289 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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132 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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220 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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+100

Meromorphic functions on $Y^2 = X^3 + 1$, genus.

Let $k$ be a field of characteristic $\neq 2$, and consider the quadratic extension $F$ of $k(X)$ generated by $\sqrt{X^3 + 1}$. What is/how do I find the genus of $F$? The progress I have so far: ...
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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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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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145 views

Is $\mathbb{Z}$ the only totally-ordered PID that is “special”?

(All my rings are commutative and unital.) Definition. Call a totally-ordered ring $R$ special iff for all non-zero $b \in R,$ every coset of $bR$ has a unique element in the interval $[0,|b|).$ ...
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123 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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66 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 ...
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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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353 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 ...
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310 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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74 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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121 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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103 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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258 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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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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286 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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413 views

Proof that a finite separable extension has only finite many intermediate fields

Let $E/F$ be finite separable extension. Is there any proof of the fact that there are only finitely many intermediate fields without using primitive element theorem or fundamental theorem of Galois ...
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90 views

$p \in C - D$, inflection point for $C$ iff inflection point for $C \cup D$.

Show that if $C$ and $D$ are projective curves in $\mathbb{P}_2$ and $p \in C - D$ then $p$ is a point of inflection for the curve $C$ if and only if $p$ is a point of inflection for the curve $C \cup ...
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93 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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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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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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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 $\langle y\rangle$ ...
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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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170 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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203 views

The order of $H$ is relatively prime to its index $[G:H]$

Suppose that a subgroup $H$ of a finite group $G$ satisfies one of the following two conditions: (i) For any nonidentity element $x$ of $H$ we have $C_{G}(x) \subset H$ (ii) If $K$ is a subgroup of ...
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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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244 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? ...
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95 views

Find the homogeneous polynomials whose set of values is closed under multiplication

Let $f(x_1,x_2,\dotsc,x_n)$ be a homogeneous polynomial. Let $$S=\{f(a_1,a_2,\dotsc,a_n)\mid a_1,a_2,\dotsc,a_n \in\Bbb Z\}.$$ If $S$ satisfies the following condition: for all $m,n\in S$, we have ...
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Ramification group fixing an unramified extension

For a Galois extension of local fields $L/K$ with Galois group $G$, define the ramification groups $G_i$ by $$G_i = \{ \sigma \in G : \nu_{L}(\sigma(x) - x) \geq i+1 \text{ } \forall x \in ...
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Orders of elements and homomorphisms.

Corollary 4.6.8 There is a group $G$ of order $n^3$ given by $G= \{b^ic^ja^k \mid 0 ≤ i, j, k < n\}$, where $a$, $b$, and $c$ all have order $n$, and $b$ commutes with $c$, $a$ commutes with ...