实封闭的场的度谱
Degree Spectra of Real Closed Fields
摘要：
一些研究人员最近建立了，对于每一个图灵度$\boldsymbol{c}$，所有$\boldsymbol{c}$ -可计算实数的实封闭的域都有谱$\{\boldsymbol{d}~:~\boldsymbol{d}'\geq\boldsymbol{c}"\}$。我们调查的频谱实封闭的领域进一步，首先集中在可计算实数领域$\mathbb{R}_{\boldsymbol{0}}$的子域，然后阿基米德实封闭的领域更普遍，最后非阿基米德实封闭的领域。对于每一个不可计算的，可计算的集合$C$，我们产生一个实封闭的$C$ -可计算的子字段的$\mathbb{R}_{\boldsymbol{0}}$没有可计算的副本。然后我们建立一个阿基米德实封闭的域，它没有可计算的副本，但有一个可计算的枚举的Dedekind削减它实现，和一个可计算表示的非阿基米德实封闭的域的剩余域没有可计算的表示。
英文摘要：
Several researchers have recently established that for every Turing degree $\boldsymbol{c}$, the real closed field of all $\boldsymbol{c}$-computable real numbers has spectrum $\{\boldsymbol{d}~:~\boldsymbol{d}'\geq\boldsymbol{c}"\}$. We investigate the spectra of real closed fields further, focusing first on subfields of the field $\mathbb{R}_{\boldsymbol{0}}$ of computable real numbers, then on archimedean real closed fields more generally, and finally on non-archimedean real closed fields. For each noncomputable, computably enumerable set $C$, we produce a real closed $C$-computable subfield of $\mathbb{R}_{\boldsymbol{0}}$ with no computable copy. Then we build an archimedean real closed field with no computable copy but with a computable enumeration of the Dedekind cuts it realizes, and a computably presentable nonarchimedean real closed field whose residue field has no computable presentation.
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