论文标题
在高斯阶乘上及其与环形元素$λ$ -INVARIANTS的连接
On Gauss factorials and their connection to the cyclotomic $λ$-invariants of imaginary quadratic fields
论文作者
论文摘要
在本文中，我们建立了高斯阶乘与伊瓦泽（Iwasawa）的环形元素$λ$ - invariant之间的联系，用于虚构的二次field $ k $。结果，我们将以$ m = 3 $和$ m = 4 $的1-概率和Dilcher之间的态度解释，以及$λ$ -Invariants for $ k = \ kathbb {q}（q}（q}（\ sqrt {-3}）$ and $ k = iS）我们将后者的素数称为``非平凡''。我们还将看到，当$ d = 2,5 $和$ 6 $时，$ k = \ mathbb {q}（\ sqrt {-d}）$也是如此。作为推论，我们发现表格$ p^2 = 3x^2 + 3x + 1 $的Primes $ p $始终是非琐事的，对于$ k = \ mathbb {q}（\ sqrt {-3}）$。最后，我们表明，$ k = \ mathbb {q}（i）$和$ k = \ mathbb {q}（\ sqrt {-3}）$的非平凡素数$ p $分别由Modulo $ p^2 $一致性涉及Euler和Glaisher数字。
In this paper we establish a connection between the Gauss factorials and Iwasawa's cyclotomic $λ$-invariant for an imaginary quadratic field $K$. As a result, we will explain a corespondance between the 1-exceptional primes of Cosgrave and Dilcher for $m = 3$ and $m = 4$, and the primes for which the $λ$-invariants for $K = \mathbb{Q}(\sqrt{-3})$ and $K = \mathbb{Q}(i)$ is greater than one, respectively. We refer to the latter primes as ``non-trivial'' for their respective fields. We will also see that similar correspondences are true for $K = \mathbb{Q}(\sqrt{-d})$ when $d = 2,5$ and $6$. As a corollary we find that primes $p$ of the form $p^2 = 3x^2 + 3x + 1$ are always non-trivial for $K = \mathbb{Q}(\sqrt{-3})$. Last, we show that the non-trivial primes $p$ for $K = \mathbb{Q}(i)$ and $K = \mathbb{Q}(\sqrt{-3})$ are characterized by modulo $p^2$ congruences involving Euler and Glaisher numbers respectively.
