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13

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 two vector spaces? Peano gave the modern axiomatization of them only in 1888, and he called them linear systems. But the use of "an algebra" in essentially ...


9

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 $Y$ depends only on (what we now call cohomology of) its fundamental group $\pi=\pi_1(Y)$: [Since] homology and cohomology groups of $Y$ (with coefficients in $A$) ...


9

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, inspired by Dedekind. It was Artin who finally detached the Galois theory from the problem of solving equations algebraically, and gave a presentation that freely ...


8

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 Kiernan's The Development of Galois Theory from Lagrange to Artin (Archive for History of Exact Sciences Vol. 8, No. 1/2, pp. 40–154).


7

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, International Journal of Algebra and Computation, 11, (3), 301-334, 2001] It was a question of M.-P. Schützenberger (MPS) to Krob and myself to prove (maybe at ...


7

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 methods of algebraic topology have invaded extensively the domain of pure algebra, and initiated a number of internal revolutions. The purpose of this book is to ...


6

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) specifies that it was coined in the paper Combinatorial analysis situs (1926). Weibel (p. 801) adds that "chain complex" was introduced by Mayer in Über abstrakte ...


6

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 theorem, some "algebraic" and some "geometric". The first geometric proof was given by Baer and Levi in 1936, and this is the proof we present. There is another ...


6

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 "isomorphic" to another group G if one can establish between their substitutions [i.e., elements] a correspondence such that: (1) each substitution of G ...


6

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 some annotated notes which make understanding the exposition easier.


6

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 cyclotomic ring that fails to have unique factorization. One of Kummer's first papers (in 1847) where he discusses how to save unique factorization in ...


6

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'schen Gleichungstheorie" Mathematische Annalen 43 (1893) 548 - 549


6

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 the Product Formula for Valuations" (Bull. AMS 51 (1945), 469-492). Here they write $S$ for a finite nonempty set of primes that includes all the archimedean ...


6

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 Maurice Boulanger et Pierre Samuel. On a présenté de courts extraits de films d’amateur tournés par Pierre Samuel, ancien membre de Bourbaki, durant divers ...


5

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 information about it is available online. A good start is Math Central where we read that "The story begins in 1602, when Nathaniel Torporley (1564-1632) began ...


5

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’ensemble des éléments normaux de $\mathrm G$ est un diviseur normal qui sera dit central de $\mathrm G$. (Jahrbuch: “Die Gesamtheit invarianter Elemente einer ...


5

Since $x^r+y^r=z^r$ can be transformed into a trinomial equation, we could give credit to Lambert, since he developed functions for it already in 1758, see for example this paper here by Corless, et.al. We quickly have $x^r*(1+(y/x)^r)=x^r*(z/x)^r$. Set $p=ln(y/x)$, $q=ln(z/x)$ and $s=e^r$. Then we get $1+s^p = s^q$. By further setting $w=s^q$ and $t=p/q$, ...


5

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. The original publication of Abel's collected works was in French. MAA hails old German translation without mentioning anything about English one, and promotes ...


4

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 Analysis Situs. "We then say that the two manifolds V and V ′ are analytic continuations of each other. In this way we can form a chain of manifolds V1, V2, ...


4

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 on its own, from there the path did not go from geometry to the group law, but the other way. In 1834 Jacobi pointed out a possible connection between cubic ...


4

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 form is a polynomial in its coefficients that remains unchanged under the transformation of said coefficients induced by the action of the special linear group ...


3

[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 sense of it with Google Translate. Edit (December 28, 2018): Lo and behold, an article in English Wikipedia. Edit (January 6, 2019): Also, a short video about ...


3

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, titled "Free products and direct products" starts with: "Let $A\times B$ be the direct (or Cartesian) product of the groups $A$ and $B$, ...


3

The explanation I learned when I first learned the term Poincare patch on AdS$_{d+1}$ in the context of AdS/CFT is the following. The metric for such a patch is \begin{equation*} ds^2 = \frac 1{z^2} \left(dz^2 + \eta_{\mu\nu} dx^\mu dx^\nu \right) \end{equation*} where $x^\mu$ are coordinates on a $d$-dimensional Minkowski space and $\eta_{\mu \nu}$ is ...


3

Between algebras and vector spaces there was (and still is, as taught in some physics courses or in those known as Vector Calculus at many American universities) the algebra of vectors developed by Oliver Heaviside to serve the needs of physics. It appeared in his paper "On the forces, stresses, and fluxes of energy in the electromagnetic field", ...


3

These videos (some of them, maybe) were on view during an exhibition about Bourbaki and his early collaborators at Bibliothèque de l'École normale supérieure. Presumably the Association des collaborateurs de N. Bourbaki still has a copy of some of these videos, I don't know if, nor how, it can share them though.


3

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 Buchberger's thesis (1965). In the 1950 paper Uber die Eliminationstheorie (Abramson's English translation) it is applied to finding bases of integrals of ...


2

The shift seems quite recent. For the "short" writing, see some "old" Algebra's textbook, like: Bartel van der Waerden, Modern Algebra, Vol.I (1949 - or.ed.1930-31): page 11 for groups and page 38 for rings; Nicolas Bourbaki, Elements of Mathematics. Algebra I: Chapters 1-3 (1998 - or.ed.1970), page 30: Definition 1. A set with an associative law of ...


2

Regarding the mathematical usage of magma, it seems that the origin is the "Bourbaki circle", but it is not so easy to trace exactly the date ... The first English translation of Nicolas Bourbaki, Algèbre: Chapitres 1 à 3 (1970) dates the first pubblication to 1943. As you can see, the definition of magma is the first one (page I-1), under the heading : 1....


2

According to Edwards, in his 1887 Fundamental Theorem of General Arithmetic Kronecker was working with polynomials with integer coefficients. He proclaimed, however, that "the so-called fundamental theorem of algebra is replaced" with the title theorem, and that he "put foundations of algebra in an entirely new form". Indeed the proof applies much more ...


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