By Harold M. Edwards
Contents and remedy are clean and extremely diverse from the normal treatments
Presents a completely optimistic model of what it ability to do algebra
The exposition is not just transparent, it truly is pleasant, philosophical, and thoughtful even to the main naive or green reader
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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.