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 \(\varDelta ^0_2\) 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.
The research of the first author is supported by RSF Grant no. 18-11-00028; he is also funded by the Russian Ministry of Education and Science (project 1.451.2016/1.4) as a federal professor in mathematics. The second author was partially supported by Grant # 581896 from the Simons Foundation, and the second and third authors were both supported by grants from the City University of New York PSC-CUNY Research Award Program. The authors wish to acknowledge useful conversations with Dr. Kenneth Kramer.
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Kalimullin, I., Miller, R., Schoutens, H. (2019). Degree Spectra for Transcendence in Fields. In: Manea, F., Martin, B., Paulusma, D., Primiero, G. (eds) Computing with Foresight and Industry. CiE 2019. Lecture Notes in Computer Science(), vol 11558. Springer, Cham. https://doi.org/10.1007/978-3-030-22996-2_18
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