# Degree spectra for transcendence in fields

## TL;DR

This paper investigates the complexity of transcendence and algebraic independence relations in computable fields, showing their degree spectra can be highly varied and include non-computable degrees, which is a novel finding in the field.

## Contribution

It demonstrates for the first time that a computable field can lack a computable transcendence basis, revealing new complexity phenomena in computable algebra.

## Key findings

- Degree spectra can be any c.e. Turing degree or above a fixed Δ^0_2 degree.
- Spectra can be characterized by the ability to enumerate a Σ^0_2 set.
- Computable fields may lack a computable transcendence basis.

## Abstract

We show that for both the unary relation of transcendence and the finitary relation of algebraic independence on a field, the degree spectra of these relations may consist of any single computably enumerable Turing degree, or of those c.e. degrees above an arbitrary fixed $\Delta^0_2$ degree. In other cases, these spectra may be characterized by the ability to enumerate an arbitrary $\Sigma^0_2$ set. This is the first proof that a computable field can fail to have a computable copy with a computable transcendence basis.