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 "...
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  • 67.1k
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 ...
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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 ...
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11 votes
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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 ...
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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 $Y$ ...
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10 votes
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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 ...
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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, ...
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8 votes
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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, ...
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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 ...
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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 ...
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  • 4,084
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 ...
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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 ...
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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 ...
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  • 1,134
6 votes
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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 ...
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  • 4,084
6 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 ...
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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,...
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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’...
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6 votes
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What were the initial applications of finite fields?

In Cox's "Galois Theory" (2nd edition) he explains in Chapter 14 why Galois created finite fields: it was to construct solvable primitive permutation groups. Solvable groups were a big deal to Galois, ...
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6 votes

When was the modern field theory approach to Galois theory developed?

A first general treatment (that is, with an abstract notion of field, which is how I understand the question) of Galois theory was given by Heinrich Weber in "Die allgemeinen Grundlagen der Galois'...
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6 votes
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Who discovered the topological proof of Nielsen-Schreier theorem?

See the end of the Wikipedia link in your first sentence. The source is Rotman's Introduction to the Theory of Groups (1995), which reads on p.383:"There are today several different proofs of this ...
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 "...
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6 votes
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Where can I find the translated manuscript of Abel?

Peter Pesic's popular math book Abel's Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability contains the author's own translation of Abel's 1824 paper as an appendix. He also has ...
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  • 300
6 votes

Where can I find the translated manuscript of Abel?

I have doubts that it was translated into English. French and German works often go untranslated because it is assumed that most English speaking mathematicians can make out enough from the original. ...
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6 votes
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On the notion of a chain (as for example in chain complex)

One learns quite a bit by searching I. M. James' History of Topology (1999): The name: Lefschetz (p. 546) confirms that The felicitous term "chain" is due to Alexander while Katz (p. 119) ...
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5 votes
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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 ...
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4 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 ...
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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,...
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4 votes

On the notion of a chain (as for example in chain complex)

I believe the term chain was introduced by Poincare as part of his second definition of the term manifold; the definition I am referring to is on pg. 24-25 of Stillwell's wonderful translation of ...
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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 ...
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  • 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 ...
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