Evaluation of the Dedekind zeta functions at s=−1s=−1 of the simplest quartic fields

In this paper, we will evaluate the values of the Dedekind zeta functions at s=−1s=−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 the 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 develop a method of computing sum of divisors function for an ideal. Finally, by combining these results, we compute the values of the Dedekind zeta functions at s=−1s=−1 of the simplest quartic fields.