论文标题
通过小规模的主要理想代表射线类组的理想类别类别
Representing ideal classes of ray class groups by product of prime ideals of small size
论文作者
论文摘要
我们证明，对于每个模量$ \ mathfrak {q} $，狭窄的射线类组的每个类$ h _ {\ mathfrak {\ mathfrak {q}}}（\ mathbf {k}）$ a nutundary number field $ \ mathbf {k} $ c {k} $包含三个无用的Prime Ideals $ \ Matherfrak $ \ pripper $ \ pripeds $ \ prippers $ \ prippers $ \ p}的产品$ \ MATHFRAK {N} \ MATHFRAK {P} \ LE（T（\ MathBf {K}）\ Mathfrak {n} \ Mathfrak {Q}）^3 $，其中$ t（\ Mathbf {k}）$是$ \ Mathbf {k Mathbf {K k Mathbf {K k Mathbf} $的明确函数。为了实现这一结果，我们首先获得了针对射线类别的尖锐的Brun-titchmarsh定理，然后为狭窄的射线类别的大型亚组提供了同样明确的改进的Brun-titchmarsh定理。在途中，我们在狭窄的射线级组的二次亚组中推断出一个明确的上限，以使其最不优势，以及最不理想的大小，它是任何给定类别的$ H_ \ Mathfrak {q} {q}（\ Mathbf {k}）$的prime乘积。
We prove that, for every modulus $\mathfrak{q}$, every class of the narrow ray class group $H_{\mathfrak{q}}(\mathbf{K})$ of an arbitrary number field $\mathbf{K}$ contains a product of three unramified prime ideals $\mathfrak{p}$ of degree one with $\mathfrak{N}\mathfrak{p}\le (t(\mathbf{K})\mathfrak{N}\mathfrak{q})^3$, where $t(\mathbf{K})$ is an explicit function of $\mathbf{K}$ described in the paper. To achieve this result, we first obtain a sharp explicit Brun-Titchmarsh Theorem for ray classes and then an equally explicit improved Brun-Titchmarsh Theorem for large subgroups of narrow ray class groups. En route, we deduce an explicit upper bound for the least prime ideal in a quadratic subgroup of a narrow ray class group and also for the size of the least ideal that is a product of degree one primes in any given class of $H_\mathfrak{q}(\mathbf{K})$.
