Editing Dirichlet character (section)

== Some notable special modules ==

* 8, the smallest module whose Dirichlet characters need more than one generator
* 13, the smallest module whose Dirichlet characters contain numbers <math>\alpha</math> such that there is no [[prime number|prime]]s p in <math>Z</math> which are still primes in <math>Z[\alpha]</math> 
* 19, the smallest module whose Dirichlet characters contain numbers whose real and imaginary parts are not [[constructible number]]s
* 24, the largest module whose Dirichlet characters are all [[real number|real]] (the Dirichlet characters of the number n are all real if and only if n is divisor of 24)
* 47, the smallest module whose Dirichlet characters contain numbers <math>\alpha</math> such that the [[Class number (number theory)|class number]] <math>h^-</math> of the [[cyclotomic field]] <math>Q(\alpha)</math> is greater than 1
* 120, the smallest module whose Dirichlet characters need more than three generators
* 149, the smallest module whose Dirichlet characters contain numbers <math>\alpha</math> such that the full [[Class number (number theory)|class number]] <math>h^- \cdot h^+</math> of the [[cyclotomic field]] <math>Q(\alpha)</math> is not [[coprime]] to the smallest number such that <math>\alpha^n=1</math> (related to [[irregular prime]])
* 240, the largest module whose Dirichlet characters are all [[Gaussian integer]]s (the Dirichlet characters of the number n are all Gaussian integers if and only if n is divisor of 240)
* 383, the smallest module whose Dirichlet characters contain numbers <math>\alpha</math> such that the [[Class number (number theory)|class number]] <math>h^+</math> of the [[cyclotomic field]] <math>Q(\alpha)</math> is greater than 1
* 504, the largest module whose Dirichlet characters are all [[Eisenstein integer]]s (the Dirichlet characters of the number n are all Eisenstein integers if and only if n is divisor of 504)
* 840, the smallest module whose Dirichlet characters need more than four generators