Primitive Recursive Ordered Fields and Some Applications

We establish primitive recursive versions of some known facts about computable ordered fields of reals and computable reals and apply them to several problems of algebra and analysis. In particular, we find a primitive recursive analogue of Ershov-Madison's theorem about the computable real closure, relate primitive recursive fields of reals to the field of primitive recursive reals, give sufficient conditions for primitive recursive root-finding and for computing spectral decompositions of matrices, which applies to computing solution operators of symmetric hyperbolic systems of partial differential equations.