I'm wondering if there is existing terminology to describe fields $F$ with the properties below. I don't have a completely precise description of the concept I have in mind, but hopefully this will be enough.

  1. Every element of $F$ requires only a finite amount of information to describe (so that it can be stored in a computer in some way).
  2. There exist (finite) algorithms which can be used implement the field operations of $F$.

For example, the field $\mathbb Q$ (and any finite extension of it) has these properties, but the fields $\mathbb R$ and $\mathbb Q_p$ (where $p$ is prime) do not. I would appreciate any help in formulating my question more precisely, and any references which might discuss this concept.

  • $\begingroup$ I think the title is a bit misleading. I thought you were after mathematics formalized using the computer. $\endgroup$ – Pedro Sánchez Terraf Mar 20 '18 at 10:53

You are looking at a computable field (if your focus is on the field), or a computable presentation of a field (if your focus is on the details of how elements and operations are coded). These objects are studied in computable structure theory.

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