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Proceedings of ASARC Workshop Daecheon & Muju 2009
(2009 / vol.11 / no.2)
Evaluation of the Dedekind zeta functions at $s=-1$ of the simplest quartic fields
Hyun Kwang Kim , Jun Ho Lee
Pages. 63-79   



The simplest quartic field was introduced by M. Gras and studied by A. J. azarus. In this paper, we will evaluate the values of the Dedekind zeta functions at $s=-1$ of the simplest quartic fields. We first introduce Siegel's formula for the values of the Dedekind zeta function of a totally real number field at negative odd integers, and will apply Siegel's formula to the simplest quartic fields. In the second, we will
develop basic arithmetic properties of the simplest quartic fields which will be necessary in our computation. We will compute the discriminant, ring of integers, and different of the simplest quartic fields. In the third, we will give a full description for a Siegel lattice of the simplest quartic fields, and will develop a method of computing sum of divisor function for ideals. Finally, by combining these results,
we compute the values of the Dedekind zeta function at $s=-1$ of the simplest
quartic fields.



1. Siegel's formula
2. The simplest quartic fields
3. Description of a Siegel lattice
4. Values of zeta functions