论文标题
Galois数字字段的欧几里得理想类别奇数字段
Euclidean ideal classes in Galois number fields of odd prime degree
论文作者
论文摘要
温伯格（Weinberger）于1972年证明，如果且仅当它是欧几里得域（Euclidean域），只要普遍的Riemann假设成立，就证明了具有单位排名至少$ 1 $的数字字段的整数环。 Lenstra扩展了欧几里得域的概念，以捕获使用有限循环类组的Dedekind域，并在此设置中证明了类似的定理。更确切地说，他表明，如果且仅当其具有欧几里得理想班级时，只要有众多的Riemann假设，则具有单位排名至少1美元的数字字段的整数阶层是循环的。本文的目的是显示以下内容。假设$ \ mathbf {k} _1 $和$ \ mathbf {k} _2 $是两个奇数prime学位的galois数字字段，带有环状级别组，而希尔伯特班级字段则超过了$ \ mathbb {q} $。如果$ \ mathbf {k} _1 \ mathbf {k} _2 $在$ \ mathbf {k} _i $上被分析，那么至少一个$ \ mathbf {k} _i $（$ i \ in \ in \ in \ {1,2 \} $）必须有一个理想的类别。
Weinberger in 1972, proved that the ring of integers of a number field with unit rank at least $1$ is a principal ideal domain if and only if it is a Euclidean domain, provided the generalised Riemann hypothesis holds. Lenstra extended the notion of Euclidean domains in order to capture Dedekind domains with finite cyclic class group and proved an analogous theorem in this setup. More precisely, he showed that the class group of the ring of integers of a number field with unit rank at least $1$ is cyclic if and only if it has a Euclidean ideal class, provided the generalised Riemann hypothesis holds. The aim of this paper is to show the following. Suppose that $\mathbf{K}_1$ and $\mathbf{K}_2$ are two Galois number fields of odd prime degree with cyclic class groups and Hilbert class fields that are abelian over $\mathbb{Q}$. If $\mathbf{K}_1\mathbf{K}_2$ is ramified over $\mathbf{K}_i$, then at least one $\mathbf{K}_i$ ($i \in \{1,2\}$) must have a Euclidean ideal class.
