Degree Spectra for Transcendence in Fields

Abstract

© 2019, Springer Nature Switzerland AG. 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 Δ20 degree. In other cases, these spectra may be characterized by the ability to enumerate an arbitrary Σ20 set. This is the first proof that a computable field can fail to have a computable copy with a computable transcendence basis.