16 - Kronecker III: factorization

Summary

The essence of a factorization tool for his model was completely clear to Kronecker, who wrote

Die im Article 1 aufgestellte Definition der Irreductibilität entbehrt so lange einer sicheren Grundlage, als nicht eine Methode angegeben ist, mittels deren bei einer bestimmten, vorgelegten Function entschieden werden kann, ob dieselbe der aufgestellten Definition gemäss irreductibel ist oder nicht. […] Desshalb soll hier eine neue Methode dargelegt werden, welche nur einfache, hier bereits verwendbare Hülfsmittel in Auspruch nimmt.

The definition of irreducibility enunciated in Article 1 lacks a firm foundation, the more so since no method is indicated by which, given a function, it can be decided whether or not its definition is irreducible. […] A new method is presented, which will simply be applied here as a tool if needed.

and presented in the following pages tools which allowed him to factorize a polynomial

in Z[X];

in k[X₁, …, X_n][X] where k is a field for which there exists an algorithm for factorizing polynomials in k[X];

in k(α)[X] where α is an algebraic extension of k, a field for which there exists an algorithm for factorizing polynomials in k[X].