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Philip Ehrlich
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Real-Closed Fields and Intermediate Value Theorem

Artin and Schreier (1926) showed that a real-closed ordered field satisfies the intermediate value theorem for polynomials of a single variable. By the early 1980s authors such as Max Dickmann and Gregory Cherlin, working on the theory of real-closed rings, routinely appealed without proof or reference to the fact that:

For an ordered field $K$, $K$ is real-closed iff $K$ satisfies the intermediate value theorem for polynomials (of a single variable) over $K$.

Since that time numerous proofs of the equivalence have appeared without reference to earlier proofs and some authors, such as van den Dries in his book on o-minimality, simply define a real-closed ordered field as one that satisfies the intermediate value theorem for polynomials (of a single variable). Was the equivalence simply a "folk theorem" by the 1980s or are there earlier published proofs? If the latter, where might one find the earliest such published proof?

Philip Ehrlich
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