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In the 1st chapter of Eisenbud's book that you had mentioned, he discusses four fundamental theorems of Hilbert that appeared in 1890 and 1893 (see p. 26): the basis theorem, the Nullstellensatz, the …

Euler's attack on the lemma was flawed precisely because of the (implicit) assumption regarding the veracity of a third powers product principle in $\mathbf Z[\sqrt{-3}]$. A close reading of Euler's …

In 1882, Dedekind and Weber developed the theory of Riemann surfaces purely algebraically, by taking as their primary object of study not Riemann surfaces, but instead function fields $K$ over $\mathb …

Here is a characterization of entropy functions due to Faddeev in 1956 (see pp. 229-231 of Faddeev's paper here if you read Russian or Chapter 1 of A. Feinstein's 1958 book Foundations of information …

In the first decades of the 20th century, $p$-adic analysis (or valuation theory more generally) was regarded by many as rather exotic. After Hensel's work there was a steady development by Strassmann …

Does acceptance of conjectures before they became theorems count? Example 1. The Artin reciprocity law. When Artin went around to other people describing what he was trying to show, nobody else belie …

Since you reach back to Euler, who proved Fermat's little theorem in the form $a^p \equiv a \bmod p$ by using induction on $a$ and the binomial theorem, I think your "Frobenius endomorphism" is the $p …

I think the Busemann-Petty problem is an example like what you're asking, although changes in opinion would be due to progress (positive and negative) rather than any heuristic analysis. A great de …

You don't need the language of group theory to talk about some aspects of groups. For example, number theorists going back to Fermat were studying the group of units mod $m$ (including things like th …

See Whitney's paper from 1935 where he defined tensor products of abelian groups. There you will find the terms natural homomorphism and (especially) natural isomorphism. Whitney makes no attempt to g …

Zariski formulated the criterion for smoothness at a point in terms of the dimension of $I_x/I_x^2$ (to use your notation) as Theorem 3.2 in the paper https://www.jstor.org/stable/pdf/2371499.pdf from …

Euler found values of the Riemann zeta-function by artful manipulations of divergent series, e.g., interpreting a function that's $(-1)^{n/2}$ at even $n > 0$ and $0$ at odd $n > 0$ as $\cos(\pi n/2)$ …

Any rational function field over a finite field has genus $0$ and class number $1$, where the class number of a function field over a finite field is the number of degree-zero elements of the divisor …

Do you insist that the formula be interpreted as the value of a residue, hence requiring that it be known that the zeta-function of every number field is meromorphic around $s = 1$? It goes back to De …

In the early 1900s, Dickson introduced what he called generalized quaternion algebras over any field $K$ of characteristic not 2. These are exactly what we'd call quaternion algebras over $K$. His def …