On the computability of ordered fields

In this paper we develop general techniques for structures of computable real numbers generated by classes of total computable (recursive) functions with special requirements on basic operations in order to investigate the following problems: whether a generated structure is a real closed field and whether there exists a computable copy of a generated structure. We prove a series of theorems that lead to the result that there are no computable copies for $\mathcal{E}^n$-computable real numbers, where $\mathcal{E}^n$ is a level in Grzegorczyk hierarchy, $n\geq 3$. We also propose a criterion of computable presentability of an archimedean ordered field.