It is shown that for any computable field $K$ and any r.e. degree $\mathbf{a}$ there is an r.e. set $A$ of degree $\mathbf{a}$ and a field $F \cong K$ with underlying set $A$ such that the field operations of $F$ (including subtraction and division) are extendible to (total) recursive functions. Further, it is shown that if $\mathbf{a}$ and $\mathbf{b}$ are r.e. degrees with $\mathbf{b} \leq \mathbf{a}$, there is a 1-1 recursive function $f : \mathbb{Q} \rightarrow \omega$ such that $f(\mathbb{Q}) \in \mathbf{a}, f(\mathbb{Z}) \in \mathbf{b}$, and the images of the field operations of $\mathbb{Q}$ under $f$ can be extended to recursive functions.