论文标题
具有奇数班级的单基领域II部分：偶数
Monogenic fields with odd class number Part II: even degree
论文作者
论文摘要
在1801年，高斯证明了有许多二次二次场，具有奇数级别的数字。我们通过表明有许多给定的偶数和签名具有奇数班级的$ s_n $字段来概括这个结果。另外，我们证明，至少有$ 4 $的任何偶数学位都有无限的领域，并且至少有一个具有每个签名单位的真实嵌入式。为此，我们使用木材的参数构成了班级组，狭窄的班级组中的平均数量2 $ torsion元素，狭窄的班级组和均匀程度的单一化场班级组（并在尾部估算上精确地有条件）。这些平均值是第一个$ p $ torsion的平均值，要计算出$ p $而不是学位的cocrime（至少在$ 3 $中），揭示了Cohen-Lenstra-Martinet-Maltinet-Malle类型的课堂群体和“ Bad”狭窄课堂组的问题。
In 1801, Gauss proved that there were infinitely many quadratic fields with odd class number. We generalise this result by showing that there are infinitely many $S_n$-fields of any given even degree and signature that have odd class number. Also, we prove that there are infinitely many fields of any even degree at least $4$ and with at least one real embedding that have units of every signature. To do so, we bound the average number of $2$-torsion elements in the class group, narrow class group, and oriented class group of monogenised fields of even degree (and compute these averages precisely conditional on a tail estimate) using a parametrisation of Wood. These averages are the first $p$-torsion averages to be calculated for $p$ not coprime to the degree (in degree at least $3$), shedding light on the question of Cohen-Lenstra-Martinet-Malle type heuristics for class groups and narrow class groups at "bad" primes.
