Let $Q$ denote the field of rational numbers. Let $K$ be a cyclic quartic extension of $Q$. It is known that there are unique integers $A$, $B$, $C$, $D$ such that $$K=Q\left(\sqrt{A(D+B\sqrt{D})}\right),$$ where $$A\text{is}\text{squarefree}\text{and}\text{odd},D=B^2+C^2\text{is}\text{squarefree},B>0,C>0,GCD(A,D)=1.$$ The conductor $f\left(K\right)$ of $K$ is $f\left(K\right)=2^l\left|A\right|D$, where $$l=\left\{\begin{array}{c}3,\text{if}D\equiv 2(mod4)\text{or}D\equiv 1(mod4),B\equiv 1(mod2),\hfill \\ 2,\text{if}D\equiv 1(mod4),B\equiv 0(mod2),A+B\equiv 3(mod4),\hfill \\ 0,\text{if}D\equiv 1(mod4),B\equiv 0(mod2),A+B\equiv 1(mod4).\hfill \end{array}\right.$$ A simple proof of this formula for $f\left(K\right)$ is given, which uses the basic properties of quartic Gauss sums.

Recently Garashuk and Lisonek evaluated Kloosterman sums K (a) modulo 4 over a finite field F3m in the case of even K (a). They posed it as an open problem to characterize elements a in F3m for which K (a) ≡ 1 (mod4) and K (a) ≡ 3 (mod4). In this paper, we will give an answer to this problem. The result allows us to count the number of elements a in F3m belonging to each of these two classes.

For a number field $K$ with ring of integers ${𝒪}_K$, we prove an analogue over finite rings of the form ${𝒪}_K/{𝒫}^m$ of the fundamental theorem on the Fourier transform of a relative invariant of prehomogeneous vector spaces, where $𝒫$ is a big enough prime ideal of ${𝒪}_K$ and $m\>1$. In the appendix, F.Sato gives an application of the Theorems 1.1, 1.3 and the Theorems A, B, C in J.Denef and A.Gyoja [Character sums associated to prehomogeneous vector spaces, Compos. Math., 113(1998), 237–346] to the functional equation of $L$-functions...

We give a geometric interpretation (and we deduce an explicit formula) for two types of exponential sums, one of which is the third moment of Kloosterman sums over ${\mathbf{F}}_q$ of type $K({\nu }^2;q)$. We establish a connection between the sums considered and the number of ${\mathbf{F}}_q$-rational points on explicit smooth projective surfaces, one of which is a $K3$ surface, whereas the other is a smooth cubic surface. As a consequence, we obtain, applying Grothendieck-Lefschetz theory, a generalized formula for the third moment of Kloosterman...

The main purpose of this paper is using the analytic method and the properties of the classical Gauss sums to study the computational problem of one kind fourth hybrid power mean of the quartic Gauss sums and Kloosterman sums, and give an exact computational formula for it.

Let $q$ be a positive integer, $\chi$ denote any Dirichlet character $modq$. For any integer $m$ with $(m,q)=1$, we define a sum $C(\chi ,k,m;q)$ analogous to high-dimensional Kloosterman sums as follows: $$C(\chi ,k,m;q)=\sum _{a_1=1}^q{}^{\text{'}}\sum _{a_2=1}^q{}^{\text{'}}\cdots \sum _{a_k=1}^q{}^{\text{'}}\chi (a_1+a_2+\cdots +a_k+m\overline{a_1{a}_2\cdots a_k}),$$ where $a·\overline{a}\equiv 1modq$. The main purpose of this paper is to use elementary methods and properties of Gauss sums to study the computational problem of the absolute value $\left|C\right(\chi ,k,m;q\left)\right|$, and give two interesting identities for it.