26 votes
Accepted

Was Kolmogorov enraged after learning about the Karatsuba multiplication algorithm?

Karatsuba's own report can be found in his 1995 paper Сложность вычислений (Complexity of computations). The phrasing he uses is "сильно взволновалo", which Google does translate as "...
Conifold's user avatar
  • 76k
13 votes
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What is the origin of "an algebra" as in vector space with multiplication?

Actually, it happened in the reverse order, algebras came first, and vector spaces only later. For the vector space story see When did people start viewing a matrix as a linear transformation between ...
Conifold's user avatar
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12 votes
Accepted

History of Direct Sums and Direct Products

"Direct sum" and "direct product" did not use to mean what they now mean in the OP sense, and even today the old usage persists. Van der Waerden in Moderne Algebra (1930-31) (the ...
Conifold's user avatar
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12 votes
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How and when did the dedicated study of locally compact groups begin?

The symbolic starting date is 1933, when Haar introduced left invariant measures and proved their existence on second-countable locally compact groups, see his Der Massbegriff in der Theorie der ...
Conifold's user avatar
  • 76k
11 votes
Accepted

History of irreducible polynomials and motivation for them

I will skip the pre-history of solving polynomial equations and factoring polynomials. Let me mention that the analogy between long division of numbers and polynomials goes back to medieval Islamic ...
Conifold's user avatar
  • 76k
11 votes
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What was the motive for inventing Gröbner bases?

Fortunately, Buchberger himself described the context of his discovery, see Historical background to Gröbner's paper by Abramson. The method, in general outline, was known to Gröbner long before the ...
Conifold's user avatar
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10 votes
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(Co)Homology: From topology to the rest of mathematics?

I'd recommend Weibel’s History of homological algebra (1999)(pdf). He describes many threads, such as roots of group cohomology in Hurewicz’s observation that cohomology of an aspherical space $Y$ ...
Francois Ziegler's user avatar
9 votes
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How did the modern understanding of Galois theory come about?

The systematic modern terminology and presentation of the Galois theory is due to Artin, a part of his joint project with Emmy Noether to reformulate the "concrete" older algebra in abstract terms, ...
Conifold's user avatar
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8 votes

What are the modern connections of the Pentagramma Mirificum studied by Gauss?

The Pentagramma Mirificum is a spherical figure formed by a series of five great circle arcs, each orthogonal to the next, and it probably does deserve a Wikipedia entry. However some elementary ...
Conifold's user avatar
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8 votes
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Did Galois make use of the concept of a basis?

No, he did not. You are used to see Galois theory from the modern point of view, developed by Emmy Noether and two of her students: van der Waerden and Emil Artin. I suggest that you read B. Melvin ...
José Carlos Santos's user avatar
8 votes
Accepted

Why do we call Chinese monoid "Chinese"? Why not "American"?

I do not have the paper "Plactic-growth-like monoids" at hand at the moment but the history is correctly rendered in [J. Cassaigne, M. Espie, F. Hivert, D. Krob, J.C. Novelli, The chinese monoid, ...
Duchamp Gérard H. E.'s user avatar
8 votes
Accepted

When did Macaulay rings become Cohen-Macaulay rings?

Actually, the name Cohen-Macaulay rings and Macaulay rings areboth due to Zariski and Samuel (Commutative Algebra, Volume 2, App. 6, p. 396, 1958): Definition 1. Let $A$ be a local ring. The common ...
user6530's user avatar
  • 3,870
7 votes
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Where are Pierre Samuel's videos of Bourbaki proceedings available?

It appears to be from the following broadcast (source): ARTE (FRANCE TÉLÉVISION - LA CINQUIÈME) Émission « Archimède » du 14 novembre 2000 consacrée à Bourbaki, réalisée avec la collaboration de ...
Uri Granta's user avatar
  • 1,204
7 votes
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Who was first to differentiate between prime and irreducible elements?

The phenomenon of nonunique factorization appears to have been first explicitly articulated in the setting of cyclotomic fields, by Kummer in the 1830s and 1840s: $\mathbf Z[\zeta_{23}]$ is the first ...
KCd's user avatar
  • 5,617
7 votes
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What was the significance of Eisenstein's discovery of invariants?

The terminology did not change that much. Algebraic forms are more often called homogeneous polynomials, but for polynomials in two variables "binary form" is still often used. Invariant of a binary ...
Conifold's user avatar
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7 votes

(Co)Homology: From topology to the rest of mathematics?

As a complement to the answer provided by Francois Ziegler, I would add the first three paragraphs of Homological Algebra (1956), by Henri Cartan and Samuel Eilenberg: During the last decade the ...
José Carlos Santos's user avatar
6 votes
Accepted

Where does the letter S in "$S$-units" and in localization $S^{-1} R$ come from?

As Francois Ziegler suggests in his comment, the notation $S$ and term $S$-unit might go back to Artin and Whaples in their paper about the product formula: "Axiomatic Characterization of Fields by ...
KCd's user avatar
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6 votes

Gauss's anticipation of quaternions and their relation to congruences

I asked a mathematician who is an expert to abstract algebra and he showed me that Gauss's congruence was actually correct. To prove Gauss congruence let's introduce the following notation: $$x = a+bi,...
user2554's user avatar
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6 votes
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Jordan called isomorphisms (iso.) and homomorphisms "iso. holoedriques" and "iso. meriedriques" respectively; translation of holoe/meried-driques?

To quote from Jordan's Traité des substitutions et des équations algébriques (1870), p. 56 (https://archive.org/stream/traitdessubstit00jordgoog#page/n79/mode/2up): "§ 67. A group Γ is called "...
Christopher K.'s user avatar
6 votes
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How did the terms "center" and "centralizer" come up in group theory?

The center (originally the central) seems to have appeared between the first and second edition of Burnside’s book (1897, §53 vs. 1911, §93) and more precisely in de Séguier (1904, §51): Ainsi l’...
Francois Ziegler's user avatar
6 votes

When did mathematicians realize that theory of algebraically closed fields admits quantifier elimination?

A useful sketch of history is given in Alfred Tarski's Elimination Theory for Real Closed Fields by van den Dries. The result was known to Tarski by 1948 (when his Decision Method for Elementary ...
Conifold's user avatar
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5 votes
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What are the modern connections of the Pentagramma Mirificum studied by Gauss?

[I do not have enough reputation to add comments so I have to make this an answer] FWIW, I have written a Polish Wikipedia article on pentagramma mirificum. For the time being, you can try to make ...
Marcin Ciura's user avatar
5 votes
Accepted

Who first identified the group structure of an elliptic curve?

In the 17th century Bachet and Fermat gave algebraic formulas for doubling a point on a cubic, and Newton showed how to do it in terms of chords and tangents. But that is as far as geometry progressed ...
Conifold's user avatar
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4 votes
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Where did the notion of the product in a category first appear?

It appears in MacLane's 1950 paper Duality for Groups, published in the Bulletin of AMS. Of course, he is dealing specifically with the category of groups, but the definition is categorical. Section 3,...
Conifold's user avatar
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4 votes

Why is the number of elements in a group called "order"?

I will promote my comment to an answer. The theory of permutations and permutation groups was the original (abstract) setting of group theory, and so the term originated there. I believe the reason ...
user1729's user avatar
  • 141
4 votes

Why was solving polynomial equations historically considered so interesting?

The conics as geometric shapes were investigated comprehensively by many ancient Greek mathematicians. After Descartes's innovation of introducing coordinates, they were seen as essential examples of ...
Mozibur Ullah's user avatar
4 votes
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Why is the term "isotropic" used to describe a quadratic form and a vector?

This is an example of how a term migrates from the original context by broken telephone through various generalizations and transfers. It started with Poncelet introducing "imaginaries", i....
Conifold's user avatar
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4 votes

First Use of the Short Exact Sequence

To narrow down your search: The lower bound is (probably) 1941. Dieudonne in his book "A history of Algebraic and Differential Topology" writes when discussing Gysin's exact sequence (which ...
Moishe Kohan's user avatar
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4 votes
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Intuitions for Frobenius' generalization of characters to nonabelian finite group given the historical context

We can’t redo the history. It just happens to be the case that Dedekind’s questions to Frobenius about group determinants were the original inspiration. It is not intuitive. Only later did Frobenius ...
KCd's user avatar
  • 5,617
3 votes
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Could a "field" have non-commutative multiplication originally?

Not quite. There was some vagueness in Dedekind's early formulations, but the tendency was to use "Körper" or "field" when the multiplication is commutative from the start. As a curiosity, in Russian ...
Conifold's user avatar
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