The abstract-algebra tag has no wiki summary.

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### Rewrite sum of radicals equation as polynomial equation

My question is about a method described in Dr.Math forum for simplifying equations involving sums of radical functions.
(The following is a transcription of the example given by Dr. Vogler):
--- ...

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233 views

### Is there any research of universal algebras axiomatized by non-Horn clauses?

A Horn clause in the language of a universal algebra is a disjunction of equations and of at most one inequality
("equation" and "inequality" are the terms used by A.Horn in his paper "On sencences ...

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215 views

### Adjunction algebra - is there anything similar to this in abstract algebra?

I call adjunction algebra a universal algebra with one binary operation denoted as the punctuation sign (;) "semicolon" (but I will be using only one space after it, not on both sides - to avoid going ...

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110 views

### Multiplicativity of the Ideal Norm

I asked this question on StackExchange twice, and was never able to get an answer. This is not quite a research-level question, but it is sufficiently difficult and interesting from a pedagogical ...

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184 views

### quasiprimitive non-solvable groups

I'm looking for the reference about quasiprimitive unsoluble groups. Actually we can find a lot of useful things about quasiprimitive solvable groups in "Representations of solvable groups by Manz and ...

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368 views

### What kind of algebra is the class of ordered pairs equipped with the binary operation which forms them?

There are many definitions of ordered pair in set theory, but all such definitions have the characteristic property of ordered pair:
$ \ \ \ \ \ \ (x, y) = (x', y') \leftrightarrow \ (x = x' \ and \ ...

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648 views

### Is there one binary operation foundational for set theory?

The membership relationship "$\epsilon$" is foundational for set theory, in the sense that the axioms of any set theory are formulated in the language of "$\epsilon$". Naturally, the question arises ...

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241 views

### If questions are formalized as ideals of a boolean algebra, what kind of algebra of questions appears from Stone representation theorem?

Affirmative propositions make up a Boolean algebra, and Boolean algebras became part of classical algebra for over one century ago - in this sense they are "simple". But I did not encounter in ...

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228 views

### Isomorphism of real closed fields

Given two real closed fields $R_1$ and $R_2$ such that both have cardinality continuum, archimedean, but not necessarily complete. Assume further that they are back and forth equivalent (in the ...

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**1**answer

409 views

### A weird question about two weird decompositions of $\mathbb{R}$ as a $\mathbb{Q}$-vector space

While working in a question about the affine group $\text{Aff}(\mathbb{R})$, I have come up with the following strange question about the real numbers:
Question: Do there exist a non-trivial ...

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**1**answer

343 views

### How does Constructive Quantum Field Theory work?

Please correct me if I'm wrong, but it seems to me that two and three dimensional axiomatic quantum field theory were constructed as follow: the wightman axioms were formulated in euclidean space via ...

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437 views

### Do all exact sequences $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split for finitely generated abelian groups?

Suppose $A$ and $B$ are finitely generated Abelian groups. Are all exact sequences of the form $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split?
If not, is there an example?
...

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433 views

### Given a formal power series ,decide whether there exists a polynomial the series satisfies and if it exists,how to write it down?

Given a formal power series $$y(x)=\sum_{i=0}^{\infty} a_i x^i$$ Is there an algorithm that decides whether there exists a polynomial$$ P(x,y)=p_n(x)y^n+p_{n-1}(x)y^{n-1}+\cdots+p_0(x)=0,p_j(x)\in ...

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410 views

### Kunneth spectral sequence

In Rotman's Homological Algebra, 1st edition, there is written:
Is every detail of 11.31-11.35 correct? Isn't the spectral sequence in 11.35 1st quadrant and not 3rd quadrant? Do 11.34-35 also ...

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108 views

### SHPS and SPHS inequality using monounary algebra

Let $A_n = \{(1,\ldots,n) , f \}$ where $f(i) = (i+1)$ if $i \neq n $ otherwise $f(n) = 1$.
This describes a mono unary algebra.
The proof for $HPS \neq SPHS$ I know uses metabelian groups and was ...

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391 views

### Divisible torsion Z-modules

I am trying to prove that for any divisible torsion $\mathbb{Z}$-module $V$,
this map
$f:\mathbb{Q}/\mathbb{Z}\otimes\text{Hom}(\mathbb{Q}/\mathbb{Z},V)\longrightarrow V$
is an isomorphism via
...

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89 views

### Positivstellensatz for non-polynomial term

Can we use Positivstellensatz (P-satz) below for a non-polynomial term?
P-satz:
Let $R$ be real closed field. Let $f,g,h$ be finite families of polynomials in $R[X_{1} ,...,X_{n}]$. Denote by P the ...

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**1**answer

250 views

### operations on ideals in a subring of number field

For three ideals $I, J$ and $K$ of a subring $R$ in a number field $L$,
does this equality hold in general?
$(I+J) \cap K = (I \cap K) + (J \cap K)$
I have no counterexample yet but I couldn't prove ...

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378 views

### Misunderstanding in the hypotheses of Schlessinger's criterion

Good day to everyone.
In studying deformation theory of Galois representations, I've come surely to an error, relating Schlessinger's criterion.
Let's fix a representation $\bar{\rho}$ of a group ...

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**1**answer

160 views

### Algebra generated by a tree [Edit] [closed]

Suppose that $(T,\leq)$ is a partially ordered set, we say $T$ is a tree* if for every $i\in T$, $\{s: s\in T, s\leq t\}$ is a well-founded chain.
What I need to know is: Can the algebra ...

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**1**answer

260 views

### Does every commutative variety of algebras have a cogenerator?

By a commutative variety $\mathcal{V}$ I mean a classical variety of algebras for some $(\Sigma,E)$, such that each pair of operations in $\Sigma$ commutes.
Equivalently (i) every interpretation of ...

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129 views

### intersection growth of free profinite groups

Let $F_n$ be a free profinite group of finite rank $n$ and let $V_k$ denote the intersection of all open subgroups of $F_n$ of rank at most $k$ ($k \in \mathbb{N}$).
My questions are:
can I ...

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123 views

### Hall's paper on the profinite groups and Andre Weils “voisinage” notion

I am reading through a classical paper A Topology for Free Groups and Related Groups
by Marshall Hall Jr. in which profinite groups are defined for the first time.
There he defines on p. 129:
...

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47 views

### Decomposition results for locally commutative semigroups

Every finite abelian group is the direct product of its cyclic groups of prime order, and every commutative monoid divides a product of its cyclic submonoids. Could these results generalized to ...

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185 views

### finite index, self-normalizing subgroup of $F_2$ [closed]

Denote $F_2=\langle a, b\rangle$ to be the free group on two generators $a, b$.
Let $H\leq F_2$ to be a subgroup with finite index $n$, so $H\cong F_{n+1}$ by Nielsen–Schreier theorem, recall that ...

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415 views

### How is a descent datum the same as a comodule structure?

For a homomorphism of commutative rings $f:R\to S$, there are at least two notions of a descent datum for this map. One of these is to be an $S$-module $M$, with an isomorphism $M\otimes_R S\cong ...

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161 views

### Abelian Subgroup in an infinite non-abelian 3-group

Does an infinite non-abelian 3-group of exponent greater than or equal to 9 has an infinite abelian subgroup?
I know that 2-groups and 3-group of exponent 3 has an infinite abelian subgroup. I wonder ...

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### What can we say about a semigroup that's acted on by an ideal of polynomials?

This got no response on MSE, so posting here.
Let $S = \{ (a,b) \in \Bbb{Z}^2 : \gcd(a,b) \neq 1 \} \cup (1,1)$. Then $S$ forms a semigroup. The operation being componentwise multiplication.
Let ...

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495 views

### Wedderburn decomposition of $D_{5}$

This is crossposted from MSE. The question:
Find the Wedderburn decomposition of $D_{5},$ the dihedral group of order 10, over the ﬁeld $\mathbb{F}_{3}.$
I have shown that the irreducible ...

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**1**answer

116 views

### Does this solution guarantee $det(A)=0$ where $A\in M(R)$? [closed]

Suppose $R$ is a commutative ring with identity $1$ and the following matrix equation holds:
$\begin{pmatrix} a_n & & \\
\vdots & \ddots & \\
a_1 ...

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97 views

### Varieties of rational languages and (pseudo-)varieties of finite monoids, question regarding closure property

Let $\mathcal Rat(A)$ denote the class of rational (or regular) languages over the alphabet $A$, a subset $\mathcal V(A) \subseteq \mathcal Rat(A)$ is called a variety of (rational) languages iff
...

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232 views

### Polynomiality of functions over residue rings

Suppose $\mathbb{Z}/m \mathbb{Z}$ is a residue ring for some $m \in \mathbb{N}$. If $m=p$ is a prime number then every function $f:\mathbb{Z}/p \mathbb{Z} \rightarrow \mathbb{Z}/p \mathbb{Z}$ is a ...

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156 views

### sublattice generated by lattice points intersecting a convex set

Suppose that $M\subseteq \mathbb{Z}^n$ is a module such that $\mathbb{Z}^n/M$ is free and $S\subseteq \mathbb{R}^n$ is a bounded, symmetric (around $0$) convex set. Let $M'$ be the module generated by ...

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383 views

### How universal is operadic approach to studying algebras?

I have just started to read about operads, so this question might be silly.
So it seems to me that any "reasonable" class of algebras can actually be defined as a class of all algebras over a certain ...

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373 views

### Generalizing detropicalization

Given an identity in max,plus arithmetic, are there ways to turn it into an ordinary algebraic identity it other than by replacing addition by multiplication and replacing max by series-plus or by ...

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639 views

### Commutative associative rational binary operations

What are all the nondegenerate rational binary operations that are commutative and associative? (Examples: $(x,y) \mapsto x+y$, $xy+x+y$, $xy/(x+y)$.)
Feel free to re-tag if you can think of ...

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254 views

### growth rate of $\mathbb{Z}^2\rtimes_{\sigma} \mathbb{Z}$?

I am interested in the growth rate of this type of group: $G=\mathbb{Z}^2\rtimes_{\sigma} \mathbb{Z}$, where $\sigma(a)=\begin{pmatrix}x&y\\z&w\end{pmatrix}\in SL_2(\mathbb{Z})$, where $a$ is ...

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152 views

### If $F(a_1,\ldots,a_k)=0$ whenever $a_1,\ldots,a_k$ are integers such that $f(x)=x^k-a_1x^{k-1}-\cdots-a_k$ is irreducible, then $F\equiv0$

I'm trying to understand a proof of the following theorem (from section II of Hall's paper An Isomorphism Between Linear Recurring Sequences and Algebraic Rings):
If $F(a_1, \ldots, a_k)$ is a ...

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148 views

### Solving a system of rational functions

Given pairwise distinct numbers $c_1, c_2, \dots c_n \in \mathbb{C} \setminus \{0\}$, does the system of equations $$\frac{6}{c_k} + \sum_{i \ne k} \frac{2}{c_k - c_i} = \sum_{i = 1}^n \frac{1}{c_k - ...

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536 views

### Shape of axioms in abstract algebra

When defining abstract algebraic structures (like monoids, groups, etc...), are there some constraints on the shape of the axioms, for the structure to have good properties that we implicitly use in ...

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383 views

### $\mathbb{Z}G$ (left) Noetherian$\Rightarrow$ $l^1(G)$ is a flat $\mathbb{Z}G$-(right) module?

Let $G$ be a countable discrete group (not necessarily abelian), and suppose the group ring $\mathbb{Z}G$ is a left-Noetherian ring, for example, when $G$ is a polycyclic-by finite group.
Denote the ...

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### Lazard's $\Gamma_n(f)$ as cocycle

In Michel Lazard's "Commutative Formal Groups" Springer Lecture Notes, he defines an operator on a polynomial 3-cochain $f$ denoted $\Gamma_n(f)$, which defines as the $n^{th}$ homogeneous piece of ...

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101 views

### “Symmetric” Polynomial 4-cocycles

It is an old theorem of Heaton's (based on work of Eilenberg and MacLane), that a polynomial 3-cocycle $f(x,y,z)$ which is "symmetric," in the sense that $f(x,y,z)-f(x,z,y)+f(z,x,y)=0$, is always a ...

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172 views

### English translation of Steinitz 1910?

Does there exist an English translation of Steinitz' 1910 work "Algebraische Theorie der Körper"?
http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002167042

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### classification of rank $2$ $\mathbb{Z}/p^n\mathbb{Z}$-algebra with invertible discriminant

Let $p$ be a prime number and $n$ be an integer. Let $A$ be an $\mathbb{Z}/p^n\mathbb{Z}$-algebra of rank $2$ whose discriminant is non invertible. In Serre's book lecture on the mordell Weil theorem ...

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179 views

### Find a special element in group algebra

Let $$G=\langle x, y, z\mid xyx^{-1}=zy, xzx^{-1}=z, yz=zy\rangle,$$ denote $l^1(G)^{\times}$ to be the set of units in $l^1(G)$, which we have considered as a ring with multiplication defined by the ...

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### Example of a ring whose minimals are annihilators of idempotents?

I'm looking for examples† of rings with the property that for each
$P={\rm Ann}_R(a)\in{\rm Min}(R)$ then $a\in R$ is idempotent (ie $a^2=a$)
† other than domains!

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121 views

### Is a certain group related to a primitive L function isomorphic to $Gal(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})$ for some $\ell$?

I define the notion of "Galois class of L functions" in the following way:
$A$ is a Galois class of L functions if and only if the follwing three conditions hold simultaneously:
1) every element ...

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869 views

### Diagonalize the simultaneous matrices and its background

For two $n \times n$ nonnegative definite Hermitian matrices $A$ and $B$ over the real number field $\mathbb R$:
Question1:Is there always a
nonsingular matrix $P$ over the same
field $F$ ...

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450 views

### What fields can be used for an inner product space?

Math people:
The title is the question: What fields can be used for an inner product space?
This question has been discussed in Math Stack Exchange with no definitive resolution. A similar question ...