Cohen-Lenstra-Martinet算法和Greenberg的$p$ -合理性猜想的数值验证
Numerical verification of the Cohen-Lenstra-Martinet heuristics and of Greenberg's $p$-rationality conjecture
摘要：
本文通过一系列的数值实验来支持Greenberg的$p$ -有理猜想，给出了一类$p$ -有理四次域，并找到了$p$ -有理多次域的新例子.在多二次和多三次域的情况下，我们表明，猜想是一个后果的Cohen-Lenstra-Martinet启发式和猜想的霍夫曼和张的$p$ -进调节器，我们带来了新的数值数据来支持这些programmures的扩展。我们比较了已知的算法工具，并提出了一些改进。
英文摘要：
In this paper we make a series of numerical experiments to support Greenberg's $p$-rationality conjecture, we present a family of $p$-rational biquadratic fields and we find new examples of $p$-rational multiquadratic fields. In the case of multiquadratic and multicubic fields we show that the conjecture is a consequence of the Cohen-Lenstra-Martinet heuristic and of the conjecture of Hofmann and Zhang on the $p$-adic regulator, and we bring new numerical data to support the extensions of these conjectures. We compare the known algorithmic tools and propose some improvements.
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