We refine known facts about primitive recursive ordered fields to Grzegorczyk’s classes \(\mathcal {E}^n\) for every \(n\geqslant 3\) , and prove some results specific for the ordered fields in these classes. In particular, we prove versions of \(\mathcal {E}^n\) -quantifier-elimination for real closed fields and of the \(\mathcal {E}^n\) -computability of the real closure of an \(\mathcal {E}^n\) -ordered field, study the splitting and root-finding properties of \(\mathcal {E}^n\) -computable ordered fields, establish relationships between \(\mathcal {E}^n\) -computable ordered fields of reals and the field of \(\mathcal {E}^n\) -computable reals, state the \(\mathcal {E}^n\) -computability of spectral decompositions, and show that the introduced hierarchies of ordered fields do not collapse.