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The ancient Greeks were quite strict in their separation of pure mathematics (mathematics) and applied mathematics (logistics). Euclid in his elements covered the basics of pure mathematics: line segm …

Hilbert and Weber were more or less working simultaneously and independently on questions that led them to introduce "class fields". Weber was interested in extending Dirichlet's theorem on primes in …

Maarten Bullynck has studied relations between Lambert's philosophical ideas and his mathematics. See http://www.kuttaka.org/~JHL/About.html for a start.

Indeed Hermite did not prove what today usually is called Hermite's theorem. Translated into modern terms, he shows that there are finitely many number fields of given degree and given discriminant. S …

Let me add a few remarks concerning 2. If $p \equiv 3 \bmod 4$, then ${\mathbb F}_p(i) = {\mathbb F}_{p^2}$. The relative norm of $x+iy$ is the product of $x+iy$ and its conjugate $x-iy$, but the latt …

Let me address your questions 1. - 4. What were the original goals of class field theory? The question is a little bit anachronistic; class field theory describes the splitting of primes in abelian …

Let $p \equiv 1 \bmod 3$ be a prime number, let $g$ be a be a primitive root modulo $p$, and $\zeta$ a primitive $p$-th root of unity. The three cubic periods are \begin{align*} \eta_0 & = \zeta + …

My guess is that whoever translated the fragments did not distinguish carefully between Gauss's own results and the comments by Schlesinger in https://archive.org/details/fragmentezurtheo00gausuoft As …

Martin Mattmüller, in his article Leonhard Euler, seine Heimatstadt und ihre Universität on Euler's hometown Basel, writes that this public talk (not a dissertation or written thesis), which Euler gav …

The first mathematician who talked about groups of points on elliptic curves (in the sense of Galois, i.e., in the modern sense of the word group) was Juel [Ueber die Parameterbestimmung von Punkten …

As Sesiano writes in his book, the first three books that once existed in Arabic translation (by Qusta ibn Luqa) are lost. But al-Karaji quoted extensively from Diophantus Book III (and gives almost a …

Artin (1923) wrote that Dedekind proved the case of this conjecture concerning pure cubic number fields. Dedekind published his article in 1900, but writes that it is a reworking of a draft he had wri …

As GH has already remarked, the same thing happened a lot later after Taniyama and Shimura asked whether elliptic curves defined over the rationals are modular. To begin with your last question, there …

KConrad's answer is correct, and the analytic class number formula is due to Dedekind. Yet the whole story is a little bit more complex and it is fair to say that Dedekind's analytic class number form …

"Proof by example" is a technique used by Euclid, who often proved results that hold e.g. for n integers in a typical case, say for 3 integers, as well as by Diophantus, who had to choose values for h …