Effective procedures in field theory

Abstract

Van der Waerden (1930 a , pp. 128- 131) has discussed the problem of carrying out certain field theoretical procedures effectively, i.e. in a finite number of steps. He defined an ‘explicitly given’ field as one whose elements are uniquely represented by distinguishable symbols with which one can perform the operations of addition, multiplication, subtraction and division in a finite number of steps. He pointed out that if a field K is explicitly given then any finite extension K' of K can be explicitly given, and that if there is a splitting algorithm for K , i.e. an effective procedure for splitting polynomials with coefficients in K into their irreducible factors in K [x], then(1) there is a splitting algorithm for K' . He observed in (1930 b ), however, that there was no general splitting algorithm applicable to all explicitly given fields K , or at least that such an algorithm would lead to a general procedure for deciding problems of the type ‘Does there exist an n such that E(n) ?’, where E is an arbitrarily given property of positive integers such that there is an algorithm for deciding for any n whether E(n) holds. In this paper we review these results in the light of the precise definition of algorithm (finite procedure) given by Church (1936), Kleene (1936) and Turing (1937) and discuss the existence of a number of field theoretical algorithms in explicit fields, and the effective construction of field extensions. We sharpen van der Waerden’s result on the non-existence of a general splitting algorithm by constructing (§7) a particular explicitly given field which has no splitting algorithm. We show (§7) that the result on the existence of a splitting algorithm for a finite extension field does not hold for inseparable extensions, i.e. we construct a particular explicitly given field K and an explicitly given inseparable algebraic extension K ( x ) such that K has a splitting algorithm but K ( x ) has not. (2) We note also (in §6) that there exist two isomorphic explicitly given fields, one of which possesses a splitting algorithm but the other of which does not. Thus the sort of properties of fields we are interested in depend not only on the abstract field but also on the particular representation chosen. It is necessary therefore to state rather carefully our definitions of explicit ring, extension ring, splitting algorithm, etc., and to introduce the concept of explicit isomorphism (3) and homomorphism. This occupies §§ 1,2 and 3. On the basis of these definitions we then discuss the existence of some fundamental field theoretical algorithms in explicit fields and their extension fields. This leads also to a classification of the types of extension fields which can be effectively constructed.

Uniform closure properties of P-computable functions

“…The second corollary deals with possibly nonuniform closure. Notice that this corollary applies event ofi elds in which arithmetic is recursive but over which polynomial factorization is undecidable [4]. It also shows that a polynomial degree bound is necessarily required.…”

“…Fifth, we may not find good interpolation points in order to produce Q 0 .I fwetry at most (d + 1) 4 points, the probability that at least (d + 1) 2 = d *points are good can be estimated likein the proof of [12], Theorem 5.1. We shall repeat the argument here.…”

Diophantine sets of polynomials over number fields

“…subsets of R k does not depend on the choice of θ. One can prove (see [7]) that every field which is finitely generated over its prime field is recursively stable. Furthermore, a recursive integral domain with a recursively stable fraction field is automatically recursively stable.…”

The complexity of computable categoricity

“…These considerations go back to the early 20th century, beginning with the work of Dehn [Deh11], Grete Herrmann [Her26], and Van der Waerden [vdW30]. Starting with Fröhlich and Shepherdson [FS56], Rabin [Rab60], Mal'cev [Mal61,Mal62] (and arguably Turing [Tur36]), the language and techniques of computability theory enable the modern precision possible in these studies. We can now calibrate the level of computability aligned to specific algorithmic questions.…”