A NOTE ON COMPUTABLE REAL FIELDS.
IOWA UNIV IOWA CITY DEPT OF MATHEMATICS
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A general problem of interest in this area is the problem of finding a complete classification for the class of computable ordered real fields. This note contains two theorems in this direction. It has been shown by Lachlan and Madison that every computable ordered arithmetically definable real field is a proper subfield of the field of recursive arithmetical real numbers. The first result in this note establishes that any real-closed computable real field is a proper subfield of the field of arithmetical reals. The second result answers a question which arises naturally from an earlier theorem of M. O. Rabin which asserts that the algebraic closure of a computable field is always computable. It is proven in this note that the real-closure of a computable field need not be computable. This result may strike one as surprising since an obvious extension of a theorem of Lachlan and Madison is that the real-closure of any computable ordered field is computable. Author
- Theoretical Mathematics