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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Were 3-dimensional split-complex numbers ever described in literature?

Basically, if you add two complex dimensions to reals, say $i$ and $j$, you automatically get a fourth dimension $ij$ because this number cannot be expressed using only the three dimensions. The ...
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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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2 answers
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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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6 votes
2 answers
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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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6 votes
2 answers
261 views

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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1 answer
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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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4 answers
211 views

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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2 answers
285 views

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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17 votes
1 answer
4k views

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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1 answer
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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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4 votes
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191 views

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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6 votes
1 answer
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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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11 votes
2 answers
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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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8 votes
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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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1 vote
1 answer
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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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6 votes
2 answers
784 views

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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1 answer
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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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5 votes
1 answer
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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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1 answer
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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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7 votes
2 answers
503 views

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 for "the ...
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4 votes
1 answer
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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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2 votes
1 answer
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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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7 votes
1 answer
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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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3 answers
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What was the significance of Eisenstein's discovery of invariants?

I am trying to decipher a portion of James Joseph Sylvester's 1869 address entitled "The Study That Knows Nothing of Observation", which, among other things, surveys the landscape of 19th century ...
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8 votes
2 answers
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(Co)Homology: From topology to the rest of mathematics?

I can appreciate how (co)homology arose in the context of topology/geometry. Trying to get a handle on the handles of spaces leads one to this idea. It's not obvious, but I can see how this would lead ...
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9 votes
0 answers
445 views

Whence “homomorphism”, “homomorphic”?

The kernel question leads to another : Today, homomorphism (resp. isomorphism) means what Jordan (1870) had called isomorphism (resp. holoedric isomorphism). How did the switch happen? “Homomorphic” ...
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3 votes
1 answer
225 views

Why is the term "kernel" used in algebra? [duplicate]

What was the motivation to use the word "kernel" in algebra to denote the set of all arguments which are mapped to the idendity element (by a homomorphism)? Who introduced it?
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4 votes
0 answers
309 views

What color were Emmy Noether's eyes? [closed]

I have reposted this question from MathOverflow because it is questionably relevant there. I am looking for a reference that definitively says what the color of Emmy Noether's eyes were. More ...
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5 votes
1 answer
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Where did the notion of the product in a category first appear?

In his book, Category Theory[1], Awodey writes the following about it: Next, we are going to see the categorical definition of a product of two objects in a category. This was first given by Mac Lane ...
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9 votes
1 answer
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Who first identified the group structure of an elliptic curve?

I find it amazing that the geometric construction that underlies the group law for elliptic curves gives rise to a group law. Q: Who was the first to identify the group law for elliptic curves and, ...
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1 vote
1 answer
437 views

Abbreviated Notation for Groups, Rings and Fields

Groups, Rings and Fields are often referred to by the set involved without mention of the operation(s). For example, the "group (G,+)" may be called the "group G". When did this practice originate ...
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1 answer
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Who was first to differentiate between prime and irreducible elements?

I recently learned about irreducible and prime elements in a commutative ring. However, my professor was not quite sure who was the first to make this distinction, or give an example of an irreducible ...
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