Class number and class group problems for some non-normal totally real cubic number fields

2001 ◽  
Vol 106 (4) ◽  
pp. 411-427 ◽  
Author(s):  
Stéphane Louboutin
Keyword(s):  
Class Number ◽  
Class Group ◽  
Number Fields ◽  
Cubic Number Fields ◽  
Totally Real

scholarly journals On -invariants attached to cyclic cubic number fields

2015 ◽  
Vol 18 (1) ◽  
pp. 684-698
Author(s):  
Daniel Delbourgo ◽  
Qin Chao
Keyword(s):  
Power Series ◽  
Zeta Function ◽  
Galois Group ◽  
Prime Number ◽  
Class Number ◽  
Class Group ◽  
Number Fields ◽  
Cubic Number Fields

We describe an algorithm for finding the coefficients of $F(X)$ modulo powers of $p$, where $p\neq 2$ is a prime number and $F(X)$ is the power series associated to the zeta function of Kubota and Leopoldt. We next calculate the 5-adic and 7-adic ${\it\lambda}$-invariants attached to those cubic extensions $K/\mathbb{Q}$ with cyclic Galois group ${\mathcal{A}}_{3}$ (up to field discriminant ${<}10^{7}$), and also tabulate the class number of $K(e^{2{\it\pi}i/p})$ for $p=5$ and $p=7$. If the ${\it\lambda}$-invariant is greater than zero, we then determine all the zeros for the corresponding branches of the $p$-adic $L$-function and deduce ${\rm\Lambda}$-monogeneity for the class group tower over the cyclotomic $\mathbb{Z}_{p}$-extension of $K$.Supplementary materials are available with this article.

scholarly journals Class-group problems for cubic number fields

1997 ◽  
Vol 23 (2) ◽  
pp. 365-378
Author(s):  
Stéphane LOUBOUTIN
Keyword(s):  
Class Group ◽  
Number Fields ◽  
Cubic Number Fields
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