Constructive Derivation of the Decomposition-Field of a Polynomial
Harry S. Vandiver · Annals of Mathematics · 1936
1. In attempting to set up a constructive theory of commutative algebra, that is, a theory in which every quantity or concept employed may be built up in a finite number of steps in terms of quantities which we agree in advance exist, one is faced with several fundamental problems at the beginning. Kronecker' gave a finite algorithm of the above mentioned character for determining if a given polynomial f(x) with coefficients in the rational field F is reducible or irreducible in the field F. At the same time he gave without proof a finite algorithm of a quite different character for determining if a polynomial +O(x, a) with coefficients in a field F[a], where a is an algebraic number, is irreducible or reducible in F[a]. Van der Waerden2 gave an argument to prove that Kronecker's algorithm actually yields the information mentioned, but in the course of it he assumes3 that +(x, a) has a decomposition into irreducible factors in F[a]. This does not satisfy the requirement that in order for us to speak of a decomposition of this type we must be able to show in advance how to obtain it in a finite number of steps. The method employed by Kronecker also assumes that we know the decomposition-field (Zerlegungsk6rper) of f(x), where f(a) = 0, so that we can speak of the conjugates of a. This prevents us from applying the algorithm to the constructive derivation of the decomposition-field of f(x), that is, if we write al, for a, we have