By M. Demazure, A. Grothendieck, M. Artin
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Additional info for Schemas en Groupes. Seminaire de Geometrie Algebrique du Bois Marie 1962/64 (SGA 3): III: Structure des Schemas en Groupes Reductifs
Sample text
The factor oif{x) mod g{y) corresponding to J^i{z^t^u) is, because in this case //i = d e g ^ i / d e g ^ = 1, the coefficient of u in {x + uyY — 2 — 181^^ _ i2u divided by g'{y), which is 2xy - 12 _ 2x2/2 - I2y _ 36x - 12^ = X - yf mod (y2 _ 18). ) In the same way, the factor of /(x) mod g{y) corresponding to ^2 is x + \y. Indeed, (x - \y) (x + \y) = x2 - ^y'^ = x^ - 2 mod [y'^ - 18), so /(x) = x^ - 2 splits mod g{y) = y'^ — IS into linear factors. ) 24 1 A Fundamental Theorem Example 3. f{x) — x^ + c i x + C2, g{y) = y'^ — cf -^ 4c2.
He might have known of Dedekind's work, but since he does not seem to have cited Dedekind, he probably discovered the theorem independently. The first publication of the theorem was by A. Kneser [36]. 32 1 A Fundamental Theorem I l i a of t h e repubhcation in Mathematische Werke]. In 1881, he had already described an algorithm for such factorizations in the following way: It can be assumed t h a t f{x) has no repeated factors, because otherwise one could free it of repeated factors by dividing it by its greatest common divisor with its derivative.
From §4 of [39]. The translation above is somewhat free, and Kronecker's notation F , 91, IH', d\'\ d\"'^ . . /, ci, C2, . . 5. 5, Kronecker changes f{x) to f{z + uy), where u is an indeterminate, in order to be sure that the polynomial to be factored does involve y. He then forms the "product of its conjugates," by which he surely means (see his §2) the norm of f{z -h uy) as a polynomial with coefficients in the root field of g{y)^ which is to say that it is plus or minus the constant term of the polynomial of which f{z-\- uy) is a root.