The impractical nature of von Schubert's Algorithm for factorization over ℤ is evident even from the example I have presented. The absence of a ‘reasonable’ factorization algorithm for polynomials over the integers was one of the major weaknesses of Kronecker's Model.

Berlekamp's Algorithm mended this flaw: in fact in 1969 Zassenhaus suggested substituting von Schubert's Algorithm with an application of Berlekamp's Algorithm and a lemma by Hensel.

Hensel's Lemma gives an algorithm which allows us to ‘lift’ a factorization over D/p to one over D/pn where D is a principal ideal domain and p ∈ D is irreducible.

Zassenhaus proposed computing a factorization of a polynomial f over D, based on a factorization algorithm over D/p, by the following approach:

factorize the image of f over D/p;

lift, via Hensel, this factorization to one over D/pn for a ‘suitably’ large n – the ‘suitability’ of n is based on the ability to recover all the coefficients of the factors of f over D – and

obtain the factors over D, by combining the ones over and checking if they divide f.

And when he had opened the second seal, I heard the second beast say, Come and see.

And there went out another horse that was red; and power was given to him that sat thereon to take peace from the earth, and that they should kill one another: and there was given unto him a great sword.

Revelations

The things depending from Jove: blood, tin, sapphire, mint, deer, eagle, dolphin.

E.C. Agrippa, De occulta philosophia

And the heart of Allah bleeds for the wound of Mostar's bridge. And his rage is upon the offenders. The names of Mark Mammon and Rambo Satan are engraved upon his heart.

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, …, Xn][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].

This preliminary chapter is just devoted to recalling the Euclidean Algorithms over a univariate polynomial ring and its elementary applications: roughly speaking they are essentially the obvious generalization of those over integers.

The fundamental tool related to the Euclidean Algorithms and to solving univariate polynomials is nothing more than the elementary Division Algorithm (Section 1.1), whose iterative application produces the Euclidean Algorithm (Section 1.2), which can be extended to prove and compute Bezout's Identity (Section 1.3).

The Division- and Euclidean Algorithms and theorems have many important consequences for solving polynomial equations: they relate roots and linear factors of a polynomial (Section 1.4) allowing them, at least, to be counted, and are the basis for the theory (not the practice) of polynomial factorization (Section 1.5).

They also have another, more important, consequence which is a crucial tool in solving: they allow a computational system to be developed within quotients of polynomial rings; the discussion of this is postponed to Section 5.1.

A direct implementation of the Euclidean Algorithm provides an unexpected phenomenon, the ‘coefficient explosion’: during the application of the Euclidean Algorithm to two polynomials whose coefficients have small size, polynomials are produced with huge coefficients, even if the final output is simply 1. Finding efficient implementations of the Euclidean Algorithm was a crucial subject of research in the early days of Computer Algebra; in Section 1.6 I will briefly discuss this phenomenon and present efficient solutions to this problem.

This chapter is mainly devoted to dealing with the deeper aspects of field extensions.

In Section 9.1 I prove the existence of a ‘universal extension field’ k of a field k, in which the polynomials in k[X] and even those in k[X] split into linear factors: this notion of algebraic closure generalizes the property of ℂ with respect to ℝ.

In Section 9.2, I discuss the argument which was only hinted at in Lemma 8.2.1, namely, the fact that a set of (not necessarily finite) generators of a field extension K ⊃ k can be reordered and separated so that there is an intermediate field Ktrasc such that K is an algebraic extension of Ktrasc, which is a purely transcendental extension of k. In so doing I introduce the notions of algebraic dependence and transcendental bases and show that it is possible to introduce the concept of degree for transcendental extensions, as we did for algebraic ones.

In Section 9.3 I describe the structure of finite extensions based on the above analysis and on the result that algebraic extensions of a field k are a purely inseparable extension of a separable extension of k.

In Section 9.4 I introduce another crucial concept, that of the universal field of a prime field k: this is a field which contains an isomorphic copy of any finite extension field K over k, i.e. a field in which all fields satisfying Kronecker's Model have a representation.

Kronecker's Model gives a powerful tool for computing, at least within the field of the algebraic complex numbers, and for solving polynomial equations there, provided we have an algorithm for factorizing polynomials over a given algebraic extension of the rationals. Such an algorithm exists, but its practical complexity is so unsatisfactory, that the solution of polynomial equations provided by Kronecker's ideas has no practical impact and the state of the art on Solving Polynomial Equation Systems was again in an impasse: as Macaulay put it. ‘the solution is only a theoretical one’…

… until in 1987, more than one hundred years after Kronecker's Grundzüge, Duval added an unexpected twist to Kronecker's proposal, showing how factorization can be easily avoided. Her proposal threw light on Kronecker's ideas, clarifying the philosophy behind them.

I will introduce Duval's idea by discussing how to represent rings explicitly.

Explicit Representation of Rings

In all the cases we have seen up to now, a ring A is effectively given by taking a set R, whose elements are in biunivocal correspondence with the elements of A, and defining in R those operations which turn R into a ring isomorphic to A.

And I saw when the Lamb opened one of the seals, and I heard, as it were the noise of thunder, one of the four beasts saying, Come and see.

And I saw, and behold a white horse: and he that sat on him had a bow; and a crown was given unto him: and he went forth conquering, and to conquer.

Revelations

The things depending from Saturn: bile, lead, onyx, asphodel, mole, hoopoe, eel.

E.C. Agrippa, De occulta phylosophia

Soon we will drink blood for wine.

Revolutionary of the Upper Rhine, Book of a hundred chapters