发表机构
Beijing Institute of Mathematical Sciences and Applications（北京数学科学与应用研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出研究任意2-幂次虚非CM域的新方法，证明次数大于16的二面体域结构，并完整分类了次数16类数1的104个虚非CM域，连同5个CM域完成该次数所有虚域分类。
AI 中文摘要
我们建立了一种研究任意2-幂次虚非CM域的新方法。特别地，我们证明对于次数大于16的域，二面体域要么位于三个虚二次域Q(√-2)、Q(√-3)、Q(√-67)之一的射线类域中，且该射线类域的公因子仅由整除2的素数支撑，并对相关指数有显式上界；要么是虚二次域的希尔伯特类域。我们还给出了次数为16、类数为1的104个虚非CM域的完整分类，提供了定义多项式、伽罗瓦群、基域和公因子的显式表格。这104个非CM域，连同Louboutin和Okazaki分类的5个CM域，完成了所有次数为16、类数为1的虚域的完整分类。对于CM域，问题归结为通过解析类数公式计算相对类数；而对于非CM域，由于缺乏指数为2的全实子域，需要进一步的方法，并导致更丰富的伽罗瓦群和分歧模式。
英文摘要
We establish a new method for the study of imaginary non-CM fields of arbitrary \(2\)-power degree and class number one. In particular, we show that for degrees greater than \(16\), dihedral fields either lie in the ray class field of one of the three imaginary quadratic fields \(\mathbb{Q}(\sqrt{-2})\), \(\mathbb{Q}(\sqrt{-3})\), \(\mathbb{Q}(\sqrt{-67})\), with the conductor of this ray class field supported only on primes dividing \(2\) and with explicit upper bounds on the relevant exponents, or are Hilbert class fields of imaginary quadratic fields. We also give a complete classification of the \(104\) imaginary non-CM fields of degree \(16\) with class number one, providing explicit tables of defining polynomials, Galois groups, base fields, and conductors. These 104 non-CM fields, together with the five CM fields classified by Louboutin and Okazaki \cite{lou7}, complete the classification of all imaginary fields of degree 16 with class number one. For CM fields, the problem reduces to computing relative class numbers via analytic class number formulae, whereas for non-CM fields the absence of a totally real subfield of index \(2\) necessitates a further approach and leads to a richer variety of Galois groups and ramification patterns.
