Real closed fields via strong Solovay reducibility

In computability theory, we naturally encounter some countable real closed fields. Notably, the set of all computable reals forms a real closed field. The main goal of this paper is to explore more deeply the relationship between nonrandomness and real closed fields by providing various classes of nonrandom reals that form real closed fields via Solovay reducibility. Solovay reducibility is a popular tool for studying the relative randomness of certain real numbers in algorithmic randomness theory. In our earlier studies, we explored the relationship between Solovay reducibility and Lipschitz functions, providing an analytical expression and facilitating connections with other concepts in analysis. We introduce a stronger version of Solovay reducibility, called strong Solovay reducibility, and present the hierarchy of Solovay reducibility variants that correspond to different levels of smoothness in real functions such as differentiable functions, Lipschitz continuous functions, and Hölder continuous functions. Using this type of reducibility, we successfully construct numerous countable real closed fields, thus establishing a significant connection between Solovay reducibility and classical algebraic structures. Furthermore, we distinguish between strong K -reducibility and strong Solovay reducibility by constructing counterexamples via the finite injury method.