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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How did Azumaya come up with the Nakayama lemma?

Thanks to Conifold and Chris Leary's comments, I learned that the Nakayama lemma was not first created by a mathematician named Nakayama, but that mathematicians named Azumaya and Krull first created ...
2 votes
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62 views

Who did introduce homomorphism concept for the first time?

I read that: "The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925). Homomorphisms of vector spaces are also ...
4 votes
1 answer
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Why did Kronecker develop the "adjoining a root" construction?

Kronecker is generally credited with the formalization of "adjoining a root to $f(x)=0$". Nowadays it is interpreted as the quotient $K[x]/(f)$, where $K$ is some appropriate algebraic structure in a ...
3 votes
1 answer
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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 ...
2 votes
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153 views

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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64 views

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 ...
7 votes
2 answers
586 views

Where can I find the translated manuscript of Abel?

I am looking for the translated manuscript of Abel where he proved the unsolvability of the quintic. Can anyone give me a pointer? I tried Google, but nothing came up.
1 vote
1 answer
120 views

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 ...
1 vote
1 answer
124 views

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 ...
9 votes
3 answers
725 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 ...
2 votes
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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 ...
1 vote
1 answer
129 views

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. ...
7 votes
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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 ...
4 votes
1 answer
183 views

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 ...
17 votes
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When did people know that all real polynomials of degree greater than 2 were reducible?

Admittedly, this may not be a research level question, but I am deeply curious about this. Let $f(x) \in \mathbb{R}[x]$, and write $d = \deg f$. It is well known that if $\deg f > 2$, then $f$ is ...
2 votes
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182 views

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 ...
2 votes
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160 views

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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4 answers
273 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 ...
8 votes
1 answer
933 views

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 ...
5 votes
1 answer
384 views

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 ...
4 votes
1 answer
88 views

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 ...
3 votes
0 answers
100 views

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 $K$?
4 votes
3 answers
270 views

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 ...
10 votes
1 answer
530 views

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 ...
3 votes
1 answer
131 views

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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1 answer
234 views

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 ...
2 votes
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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{...
6 votes
2 answers
425 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 ...
1 vote
2 answers
197 views

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,\...
7 votes
2 answers
215 views

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 ...
-1 votes
1 answer
245 views

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 ...
5 votes
1 answer
218 views

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, ...
9 votes
2 answers
409 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 ...
5 votes
1 answer
154 views

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 ...
1 vote
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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 ...
7 votes
1 answer
331 views

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&...
2 votes
1 answer
259 views

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 ...
5 votes
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203 views

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....
3 votes
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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 ...
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 ...
4 votes
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261 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 ...
10 votes
1 answer
385 views

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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89 views

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 ...
11 votes
2 answers
202 views

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 ...
3 votes
1 answer
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Jordan called isomorphisms (iso.) and homomorphisms "iso. holoedriques" and "iso. meriedriques" respectively; translation of holoe/meried-driques?

Stillwell mentions in his Elements of Algebra: The first to use the term "isomorphism" was Jordan, in his Traite des Substitutions [1870], the first textbook on group theory...Jordan used the word "...
3 votes
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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'?
10 votes
1 answer
444 views

What were the initial applications of finite fields?

Finite fields, I believe, were introduced by Galois in his paper "Sur la théorie des nombres", found on page 398 of his Oeuvres. In this paper Galois introduces the idea of taking a polynomial ...
1 vote
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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 ...
6 votes
1 answer
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Sum of like powers in real numbers

When and who was the first person to discover a correct formula for the real number $r$ in terms of any given three positive distinct integers $x<y<z$ such that $$x^r + y^r = z^r\,.$$ The ...