Countable Valued Fields in Weak Subsystems of Second-Order Arithmetic

This paper is part of the program of reverse mathematics. We assume the reader is familiar with this program as well as with RCA,, and WKL,, the two weak subsystems of second-order arithmetic we are going to work with here. (If not, a good place to start is [2].) In [2], [3], [4], many well-known theorems about countable rings, countable fields, etc. were studied in the context of reverse mathematics. For example, in [2], it was shown that, over the weak base theory RC&, the statement that every countable commutative ring has a prime ideal is equivalent to weak Konig’s Lemma, i.e. the statement that every infinite (0, l} tree has a path. Our main result in this paper is that, over RC&, Weak Konig’s Lemma is equivalent to the theorem on extension of valuations for countable fields. The statement of this theorem is as follows: “Given a monomorphism of countable fields h : L+ K and a valuation ring R of L, there exists a valuation ring V of K such that h-‘(V) = R.” In [5], Smith produces a recursive valued field (F, R) with a recursive algebraic closure P such that R does not extend to a recursive valuation ring R of F. However, there is little or no overlap between the contents of the present paper and [5].