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Kronecker is generally credited with the formalization of "adjoining a root to $f(x)=0$". Nowadays it is interpreted as the quotient $K[x]/(f)$, where $K$ is some appropriate algebraic structure in a wide range of structures. But if $K=\mathbb Q$ or $\mathbb Z$, some of the simplest examples possible, then the construction appears unnecessary, since our imaginary root is simply an element of $\mathbb C$, and there is no need to construct anything (assuming one is already satisfied that $\mathbb C$ exists).

Thus, for which structure(s) $K$ did Kronecker (or whoever actually first thought of this construction) develop his original construction, and why did he feel it was necessary?

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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 generally, providing that any polynomial can be split into linear factors in some extension of its field of coefficients. This is now known as the Kronecker's theorem, or the splitting field theorem, and that the proof does not depend on any analytic properties of $\mathbb C$ held great appeal in Kronecker's eyes. He names as an inspiration his "sharp critique of the method of defining quantities that began with Heine", a stand in for transcendental arguments of Cantor, Dedekind and Weierstrass.

Kronecker did not however come up with the modern quotient ring construction, being an early constructivist he had philosophical objections to Dedekind's ideals themselves, and even offered divisors as an alternative to them earlier. Instead, he formalized a more "explicit" construction of Galois, who called it "adjoining a root", arguably completing the long process of proving the FTA "algebraically". Gauss already pointed out the circularity of reasoning that roots are elements of $\mathbb C$ in Euler's and Lagrange's attempted proofs of FTA. They needed to perform computations with roots, which required treating them as elements of $\mathbb C$, in the course of proving that they are elements of $\mathbb C$. One either has to presuppose that the roots exist, which is circular, or to use continuity arguments to prove existence, which is "non-constructive".

Galois still tacitly assumed that computations with roots were valid, but he usually expressed the adjoined roots as polynomials with rational coefficients in a new indeterminate $t$, specifying $t$ to be a root of a known irreducible polynomial. Kronecker called this construction "the precious Galois principle", and proved that such presentation is always possible for polynomials with integer coefficients. Namely, given a polynomial $f(x)$ it is possible to find a polynomial $G(t)$ such that adjoining its root allows to express all roots of $f(x)$ constructively as polynomials in $t$. Thus, adjoining a single element splits $f(x)$ into linear factors, and computations with roots become justified.

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  • $\begingroup$ So are you saying that Kronecker introduced the construction because he wasn't satisfied with contemporary constructions of $\mathbb C$ ? $\endgroup$ – Jack M Apr 29 '15 at 2:18
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    $\begingroup$ @Jack M From Kronecker's 1886 letter to Mittag-Leffler:"I have put the foundations of algebra in an entirely new form... I owe this beautiful and sure foundation of algebra to my sharp critique of the method of defining quantities that began with Heine, and to the precious Galois principle". Kronecker's point seems to be that whichever way $\mathbb{C}$ is originally constructed the splitting field has to be constructed independently. Then the roots can be legitimately manipulated, and proved to belong to $\mathbb{C}$. Otherwise, we have to presuppose that the roots exist, which is circular. $\endgroup$ – Conifold Apr 29 '15 at 3:08

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