Questions tagged [abstract-algebra]
For questions about the mathematical field abstract algebra that studies algebraic structures, most notably groups, rings and fields.
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Group theory in non-European/subaltern cultures?
I'm doing undergraduate research on the history of abstract algebra (specifically permutation groups) and the notion of symmetric groups in indigenous artwork has come up several times. Is anyone ...
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Finite fields as quotients
Although finite fields are usually introduced as field extensions of fields of prime order, they also arise as quotients of number rings; e.g., $GF(9)$ comes from taking the Gaussian integers mod 3 ...
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Error-correcting codes based on Galois fields
I seem to recall reading that some French mathematician (perhaps a member of Bourbaki?) came up with the idea of basing error-correcting codes on Galois fields quite early in the development of ...
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When did mathematicians realize that theory of algebraically closed fields admits quantifier elimination?
A nice property of algebraically closed fields is that the theory that describes them ($ACF$) admits quantifier elimination: any statement can be shown equivalent (in the theory) to another statement ...
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List of textbooks on Abstract Algebra in the order of time
I am knowing Abstract Algebra things; I am searching aims of Abstract Algebra and origins of parts of Abstract Algebra. I thought original initial textbooks have explicit links to aims and origins of ...
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History behind Serre's conditions $\mathrm{S}_k$ and $\mathrm{R}_k$ for a commutative Noetherian ring
In 033Q we find defined what some sources call “Serre's conditions $\mathrm{S}_k$ and $\mathrm{R}_k$” (if you don't know what a scheme is, you can read the definition for a commutative Noetherian ring ...
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When did Macaulay rings become Cohen-Macaulay rings?
In his book on commutative rings (published 1970), Kaplansky talks about Macaulay rings. In the mid 1970's, I learned some commutative algebra from a student of his, who referred to these rings as ...
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Where did the index of a subgroup notation $[G:H]$ begin to be used?
In texts of algebra, the cardinality of cosets is written in $[G:H]$ or $|G:H|$. Where did this notation originate?
The history about $G/H$ can be found here. $[G:H]$ is called index of a subgroup. ...
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History of group actions as their own structures
I'm interested in when (and how) the modern idea of a group action developed and how group actions became their own algebraic structures.
As far as I can tell in the 19th century group actions were ...
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How and when did the dedicated study of locally compact groups begin?
How and when did the dedicated study of locally compact groups begin?
Specific instances from literature, recorded stories, etc., may help supplement the answers. There seems to be no reason why I ...
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Intuitions for Frobenius' generalization of characters to nonabelian finite group given the historical context
I'm reading about the history of character theory of finite group, especially about the invention of character theory by Frobenius.
According to most of the related papers (e.g. Pioneers of ...
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Why is the ring of algebraic integers denoted by $\mathcal O_K$?
Why/when was the curly-O notation chosen for the ring of integers of an algebraic number field $K$?
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Is there a translation of Kronecker's original work on adjoining a root of a polynomial to a field?
I would be interested in reading how Kronecker formally approached this construction, using the mathematical ideas of his time, and possibly some insight as to what he considered its philosophical ...
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The Picard Group: Origin and History
I've come to the notion of the Picard Group.
I was recently linked to this paper paper, which contains the line:
The problem of computing the Picard groups of surfaces $S \subset \mathbb{P}_{\mathbb{...
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Why is it called a group action?
A group action has two laws which roughly correspond to associativity and identity
$
\phi : (G : \textrm{Group}) \times (S : \textrm{Set}) \rightarrow S \\
\forall a, b : G . \forall c : S. \phi(a,\...
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A Survey of Modern Algebra, 1st Edition (Birkhoff and Mac Lane): The Direct Product
Background information:
I recently asked a question about the history of the concepts of the direct sum and the tensor product in group and module theory, and was given a very concise and thorough ...
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First Use of the Short Exact Sequence
A question I've been curious about for a long period of time and tried to find the answer to myself a number of times (but apparently never been able to figure out just the right thing to type into ...
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History of Direct Sums and Direct Products
So I like to get down into the details of how certain mathematical concepts came to be, and purely as a matter of curiosity, I was wondering if anyone know which mathematician first gave the ...
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How long have all the mathematicians working in the respective fields known the theory of categories
Vague the question: how long have all the mathematicians working in the respective fields known the theory of categories?
More specific questions:
Is it true that all modern working algebraic ...
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What set of criteria led Hamilton to discover the quaternions?
Frobenius's theorem states that the only finite-dimensional, associative division algebras over $\mathbb R$ are: $\mathbb R, \mathbb C, \mathbb H$ (where the last of these are the quaternions). So one ...
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Why is the term "isotropic" used to describe a quadratic form and a vector?
A quadratic form $q$ on a vector space $V$ is isotropic if $q(v) = 0$ admits a nonzero solution. What was this terminology originally intended to evoke?
There is some prior discussion on Math SE, ...
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Why was solving polynomial equations historically considered so interesting?
From reading a few accounts of the unsolvability of the quintic, I am told that, e.g., there were public contests in which people competed to solve polynomial equations, and that over the course of ...
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Who first spoke of Euclidean domains?
I am looking at Euclidean domains for the first time and it is a subject that has caught my attention. I wonder, who was the first to talk about this? I've been reading a little and so far the oldest ...
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Why is the number of elements in a group called "order"?
This is a question that I have for a long time, Maybe it is something silly, but I really want to know. Why is the number of elements in a group called "order"? I mean, the word "order&...
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How did Hamilton conclude the quaternions had to be four dimensional?
I have seen many times before that Hamilton started off believing he would need a three-dimensional system over the reals in order to describe 3D rotations. He considered numbers of the form $a + bi + ...
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Abel's "solution" of the quintic
Before Abel embarked to proving the insolubility of the quintic with radicals, he thought he had found a solution and sent his work in a letter to the Danish mathematician Carl Ferdinand Degen in 1821....
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Origin of Lang's proof of the Cayley-Hamilton theorem
Is the proof of the Cayley-Hamilton theorem given by Serge Lang in Algebra (page 561) an original one, or has it been borrowed from some earlier sources? Who came up with it first?
(Lang's proof is ...
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History of irreducible polynomials and motivation for them
I've been thinking about the history of the irreducible polynomials and why they were introduced. I found What is the origin of polynomials and notation for them?, but it's about polynomials in ...
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What is the earliest article in which Leibniz used 'matrices'?
The Chinese were using matrices ( fengcheng in the Nine Chapters of the Mathematical Art), long before they were used in Europe which suggests that possibly they were introduced by way of them. For ...
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Was Kolmogorov enraged after learning about the Karatsuba multiplication algorithm?
Some years ago, I read that Kolmogorov was so enraged that Karatsuba disproved one of his conjectures that he terminated his seminar shortly thereafter.
This Wikipedia page claims that Kolmogorov was ...
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How did Ruffini manage to extend the methods of Lagrange in order to "prove" the insolvability of the general quintic equation?
Since Lagrange published his Reflections papers during the early 1770s — around 30 years before Ruffini took up and extended the subject — I was wondering if there were any results that were ...
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Could a "field" have non-commutative multiplication originally?
Today, when the term "field" is defined in algebra, it is almost always stipulated that all fields are commutative. However, the author of these lectures says that this has not always been ...
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Origin of the term 'index of a subgroup'
The index of a subgroup $H$ in a group $G$ is the number of distinct cosets of $H$ in $G$.
Why did someone decide to call this an 'index'?
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Why did Jordan introduce his canonical form?
Camille Jordan's famous canonical form for matrices over algebraically closed fields, is considered an important result nowadays, commonly taught to all students of mathematics in undergraduate linear ...
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How did Cramer's work on permutations influence early pre-group theory (particularly the work of Lagrange and Vandermonde)?
In his Réflexions paper, Lagrange is using one of Cramer's results on the Elimination Theory for the proof of a theorem. It seems that Cramer did some early work on the permutations of variables in ...
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What was the motive for inventing Gröbner bases?
How did professor Buchberger discover Gröbner (Groebner) bases for polynomial ideals? What was the problem(s) that lead to such a discovery?
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History of Path algebras
I want some references that point the inventor of Path algebras and history/evolution of these algebras from the first idea. If possible.
I tried to search in many different places, but all times, ...
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Where are Pierre Samuel's videos of Bourbaki proceedings available?
Wikipedia's article on Pierre Samuel claims (uncitedly):
He was a member of the Bourbaki group, and filmed some of their meetings. A French television documentary on Bourbaki broadcast some of this ...
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Who first defined polynomials as sequences?
Question 1. When did the modern definition of a polynomial (as a sequence of coefficients, with multiplication defined by $\left(ab\right)_n = \sum\limits_{k=0}^n a_k b_{n-k}$) emerge?
Let me clarify:...
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Is there any relation of the word "normal" with a subgroup being normal?
From Gallian, Contemporary Abstract Algebra:
...if G is a group and H is a subgroup of G, it is
not always true that aH = Ha for all a in G. There are certain situations
where this does hold, ...
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How did the proofs of uniqueness of additive inverses originate historically?
I have encountered various abstract algebra resources that prove the impossibility of number systems with plural additive inverses for a given element, generally through the substitution property of ...
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How did the modern understanding of Galois theory come about?
The "modern" understanding of the Galois group of a polynomial is as automorphisms of the splitting field of the polynomial which keep the base field fixed. These concepts were unknown to Galois, who ...
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Did Galois make use of the concept of a basis?
I've been reading Galois' First Memoir, where he introduces Galois Theory by giving a sufficient and necessary condition for a polynomial to be solvable by radicals. The proofs are a bit sketchy and ...
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Why do we call Chinese monoid "Chinese"? Why not "American"?
Why do we call Chinese monoid "Chinese"? Why not "American"?
You can find the definition of Chinese monoid from Wikipedia. https://en.wikipedia.org/wiki/Chinese_monoid
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How did the terms "center" and "centralizer" come up in group theory?
Usually the word center means the center of a circle. I have encountered the word center in group theory, but do not see any connection with the center of a circle. I think the history of group theory ...
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What are the modern connections of the Pentagramma Mirificum studied by Gauss?
In the last years, I read a lot about a mathematical object that was discovered by John Napier in 1620 and explored much more deeply by Gauss, who called this "Pentagramma Mirificum" (latin ...
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Gauss's anticipation of quaternions and their relation to congruences
Recently i read the article "Hamilton, Rodrigues, Gauss, Quaternions and Rotations: A Historical Reassessment", which can be found freely on the internet. This article is by far the most comprehensive ...
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History of group theory character tables (as used in physics and chemistry)
Does anyone know who started using the symbols A, B, E, T (First column, left) for showing irreducible representations of symmetry groups? In older maths books I see capital gamma. Herein A= totally ...
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Notation $n=efr$ in algebraic number theory
When $\Bbb Q \subset K$ is a field extension of finite degree and when $p \in \Bbb Z$ is a prime number, the ideal $p O_K$ decomposes uniquely as a product $\prod_{i=1}^r P_i^{e_i}$ of prime ideals of ...
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Where does the letter S in "$S$-units" and in localization $S^{-1} R$ come from?
In number theory, we may encounter the notion of $S$-unit, $S$-integer, etc. where $S$ is a finite set of prime numbers (for simplicity). For instance, if $S = \{2,3\}$ then the $S$-integers are the ...
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