On the tame kernels of imaginary cyclic quartic fields with class number one

TL;DR

This paper develops a computational framework to determine the tame kernel of imaginary cyclic quartic fields with large discriminants, successfully computing cases with discriminants up to 25000.

Contribution

It introduces a novel computational architecture combining PARI, C++, parallel programming, and object-oriented design for tame kernel calculations in complex number fields.

Findings

Proved the tame kernel is trivial for specific fields with given parameters.

Established the computational approach's effectiveness for fields with discriminant less than 25000.

Demonstrated the architecture's scalability and applicability to large discriminant fields.

Abstract

Tate first proposed a method to determine $K_2\mathcal{O}_F,$ the tame kernel of $F,$ and gave the concrete computations for some special quadratic fields with small discriminant. After that, many examples for quadratic fields with larger discriminants are given, and similar works also have been done for cubic fields and for some special quartic fields with discriminants not large. In the present paper, we investigate the case of more general imaginary cyclic quartic field $F=\mathbb{Q}\Big(\sqrt{-(D+B\sqrt{D})}\Big)$ with class number one and large discriminants. The key problem is how to decrease the huge theoretical bound appearing in the computation to a manageable one and the main difficulty is how to deal with the large-scale data emerged in the process of computation. To solve this problem we have established a general architecture for the computation, in particular we have done the works: (1) the PARI's functions are invoked in C++ codes; (2) the parallel programming approach is used in C++ codes; (3) in the design of algorithms and codes, the object-oriented viewpoint is used, so an extensible program is obtained. As an application of our program, we prove that $K_2\mathcal{O}_F$ is trivial in the following three cases: $B=1,D=2$ or $B=2, D=13$ or $B=2, D=29.$ In the last case, the discriminant of $F$ is 24389, hence, we can claim that our architecture also works for the computation of the tame kernel of a number field with discriminant less than 25000.