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One landmark could be the 1693 paper "An Instance of the Excellence of the Modern ALGEBRA, in the Resolution of the Problem of finding the Foci of Optick Glasses universally"(alt. link), where Edmond …

Most ancient: Wikipedia has a daguerreotype of Gauss (1777–1855) on his deathbed. Or possibly Farkas Bolyai (1775–1856) in what look like similar circumstances. Less ancient, but allegedly photographe …

I think one reason JSTOR doesn't have “loss of generality” before 1831 is that fewer scientists wrote in English. But one finds (with minor variants merged, translations *starred, and year first publi …

$P=$ Calabi’s conjecture. Specifically, the link says “By the late 1960s, many were doubtful of the Calabi conjecture”, then Yau did “produce a "counterexample" to the conjecture. The "counterexample" …

$P=$ Carleson’s theorem, from his Abel interview: Abel first thought that he had solved the general quintic by radicals. Then he found a mistake and subsequently he proved that it was impossible t …

From this, it sounds like a very early instance is in Ascoli's proof of his theorem: pp. 545-549 of Le curve limite di una varietà data di curve, Atti Accad. Lincei 18 (1884) 521-586. (Which, alas, I …

The book is volume 46, no. 1 of the Danish Academy’s Matematisk-fysiske Meddelelser, and as such is freely available in pdf at University of Southern Denmark. (The Bibliothèque Nationale also has a …

Of course, every finite group of odd order is solvable.

Georg Frobenius, Über die Primfactoren der Gruppendeterminante, Sitzungsber. Akad. Berlin (1896) 1343-1382. The theorem is announced at the beginning, p. 1344: Der Grad $f$ ist ein Divisor der Ord …

To Carlo Beenakker's references about total differentials (which came first), one might add a word about Alexis Fontaine, inventor of the notation (notion?) of partial derivative: Fontaine, Alexis L …

I believe the answer to the question in the title is probably Cauchy, who in Méthode simple et générale pour la détermination numérique des coefficients que renferme le développement de la fonction pe …

The word at least, seems to originate with Pontryagin (1931, p. 186): 28) Wenn eine Gruppe $A$ auf eine Gruppe $B$ homomorph abgebildet ist, so heißt die Untergruppe von $A$, die aus allen Element …

I don’t know about wrong results, but compare Godement’s Algèbre (1966, and still 1980): on dit que l’ensemble $x\mathrm H$ est une classe à droite modulo $\mathrm H$ (translation: “we say tha …

This is traced in Hasse (1967) and Hawkins (2013), who writes on p. 326: According to Miyake (1989, p. 347), Hasse introduced the term “Frobenius substitution” in (1926-1930), apparently unaware …

This must come from the name Weierstrass normal form given to the elliptic integrals $$ \int\frac{ds}{\sqrt{4s^3-g_2s-g_3}},\quad \int\frac{s\,ds}{\sqrt{4s^3-g_2s-g_3}},\quad \int\frac{ds}{(s-\alpha)\ …