Degree Spectra for Transcendence in Fields
COMPUTING WITH FORESIGHT AND INDUSTRY, CIE 2019(2019)
摘要
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.
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关键词
Computability,Computable structure theory,Degree spectrum,Field,Transcendence basis
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