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I would be interested in reading how Kronecker formally approached this construction, using the mathematical ideas of his time, and possibly some insight as to what he considered its philosophical implications to be. I am referring specifically to the construction which, in modern terminology, adjoins a root of an irreducible polynomial $f(x) \in F[x]$ to $F$ by constructing the quotient $F[x]/(f(x)) $, where $(f(x))$ is the ideal of $F[x]$ generated by $f(x)$.

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    $\begingroup$ A quick skim through section 12.3, "Kronecker", of David A. Cox, Galois Theory (2nd ed. 2012) suggests that this could be quite a big subject! For details, Cox refers to a number of essays by H. M. Edwards. Perhaps these could be of some help, even if there is no English translation of Ein Fundamentalsatz der allgemeinen Arithmetik (1887, in Werke, vol. III, pp. 209--240), or earlier papers on which that one presumably depends. $\endgroup$ Commented Feb 9, 2022 at 16:34

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At least as a placeholder answer: it is my impression that that idea would have been extremely novel for its time, since there were not abstract ideas of rings and ideals and such. On the other hand, the seemingly-tangible idea that a (reasonable) field extension of $\mathbb Q$ is obtained by adjoining suitable complex numbers (which we believe exist, etc.), had serious limitations of scale.

These limitations can be seen in A. Weil's "Foundations of Alg Geom", in the 1940s, which needed algebraically closed field extensions of uncountable transcendence degree. (True, the seeming need for "Grothendieck universes" in the most-cavalier development of modern algebraic geometry has some serious foundational stuff going on, too.)

But, no, I don't know of a translation.

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    $\begingroup$ It probably is like Cauchy’s earlier proposed definition of $\mathbf C$ as congruence classes of real polynomials modulo $x^2+1$. He would not have used cosets, but just remainders with addition and (especially) multiplication done modulo $x^2+1$. Galois probably did something similar to build finite fields of nonprime order. There must be some specific paper of Kronecker where the idea was presented. $\endgroup$
    – KCd
    Commented Feb 9, 2022 at 3:20
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    $\begingroup$ Thank you for your answer! In this answer, someone explained that Kronecker didn't like Dedekind's notion of ideals so I imagine his construction was very different than what we'd see in a modern algebra book. This is partly why I'd like to see his original construction. $\endgroup$ Commented Feb 11, 2022 at 22:30
  • $\begingroup$ @KCd, I wasn't aware that Cauchy had proposed such an idea! Do you know where I could read some more about that? $\endgroup$ Commented Feb 11, 2022 at 22:32
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    $\begingroup$ @PrimeNumbers this was proposed by Cauchy in 1847. See page 63 (Chapter 3) of the book "Numbers" by Ebbinghaus et al. $\endgroup$
    – KCd
    Commented Feb 11, 2022 at 23:02

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