 
Summary: A Constructive Version of the
Hilbert Basis Theorem
Aaron Hertz
May 7, 2004
1 Introduction
The Hilbert Basis Theorem was the first major example of a nonconstructive
proof recognized in mathematics. Gordan said, on the subject of the theo
rem, "das ist keine Mathematik, das ist Theologie!"  "this is not Math
ematics, this is Theology!" [8] Although there are several equivalent state
ments of the theorem, in this paper we will consider the version which states,
in essence, that all rings of polynomials over countable fields are finitely gen
erated. (More generally, the theorem holds for polynomial ideals over any
Notherian ring. All the proofs in this paper can easily be adapted to this
more general situation.)
In this paper, we will consider two different constructive proofs. Each
is accomplished by applying GĻodel's Dialectica Interpretation to a classical
proof of the theorem. Both yield algorithms that are instances of primi
tive recursive functionals of finite types, essentially a simple programming
language in which one can only express total functions. The first, from a
standard proof, yields a constructive version requiring highertype primitive
