Degree Spectra for Transcendence in Fields

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 \(\varDelta ^0_2\) degree. In other cases, these spectra may be characterized by the ability to enumerate an arbitrary \(\varSigma ^0_2\) set. This is the first proof that a computable field can fail to have a computable copy with a computable transcendence basis.