论文标题
具有有限特征的有限残留场的多项式Dedekind域
Polynomial Dedekind domains with finite residue fields of prime characteristic
论文作者
论文摘要
我们表明，每个dedekind域$ r $位于多项式环$ \ mathbb z [x] $和$ \ mathbb q [x] $的属性中，其主要特征的残留场是有限的领域等于integer-nimized firate in integer-pliper $ p $ e $ p \ in math $ e a math $ e a math $ e a math $ e a math $绝对积分闭合$ \ overline {\ mathbb z_p} $的绝对积分闭合$ \ mathbb q $的先验元素的$ p $ -adic整数的戒指，以至于$ r = \ {f \ in \ in \ mathbb q [x] } p \ in \ mathbb z \} $。此外，我们证明了$ r $的班级组是有限生成的阿贝尔小组的直接总和。相反，此类的任何组都是$ \ Mathbb z [x] $和$ \ Mathbb Q [x] $之间的Dedekind域$ r $的类组。
We show that every Dedekind domain $R$ lying between the polynomial rings $\mathbb Z[X]$ and $\mathbb Q[X]$ with the property that its residue fields of prime characteristic are finite fields is equal to a generalized ring of integer-valued polynomials, that is, for each prime $p\in\mathbb Z$ there exists a finite subset $E_p$ of transcendental elements over $\mathbb Q$ in the absolute integral closure $\overline{\mathbb Z_p}$ of the ring of $p$-adic integers such that $R=\{f\in\mathbb Q[X]\mid f(E_p)\subseteq \overline{\mathbb Z_p}, \forall \text{ prime }p\in\mathbb Z\}$. Moreover, we prove that the class group of $R$ is isomorphic to a direct sum of a countable family of finitely generated abelian groups. Conversely, any group of this kind is the class group of a Dedekind domain $R$ between $\mathbb Z[X]$ and $\mathbb Q[X]$.
