On the computability of ordered fields

In this paper we develop general techniques for classes of computable real numbers generated by subsets of total computable (recursive functions) with special restrictions on basic operations in order to investigate the following problems: whether a generated class is a real closed field and whether there exists a computable presentation of a generated class. We prove a series of theorems that lead to the result that there are no computable presentations neither for polynomial time computable no even for $E_n$-computable real numbers, where $E_n$ is a level in Grzegorczyk hierarchy, $n \geq 2$. We also propose a criterion of computable presentability of an archimedean ordered field.