Let $K=\mathbb{Q}\left(\theta \right)$ be a pure cubic field, with ${\theta }^3=D$, where $D$ is a cube-free integer. We will determine the reduced ideals of the order $𝒪=\mathbb{Z}\left[\theta \right]$ of $K$ which coincides with the maximal order of $K$ in the case where $D$ is square-free and $\neg \equiv \pm 1(mod9)$.

We consider Weil sums of binomials of the form $W_{F,d}\left(a\right)={\sum }_{x\in F}\psi (x^d-ax)$, where F is a finite field, ψ: F → ℂ is the canonical additive character, $gcd(d,|F^{\times }\left|\right)=1$, and $a\in F^{\times }$. If we fix F and d, and examine the values of $W_{F,d}\left(a\right)$ as a runs through $F^{\times }$, we always obtain at least three distinct values unless d is degenerate (a power of the characteristic of F modulo $|F^{\times }|$). Choices of F and d for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if F is a field of order 3ⁿ with n...

Let $L=K\left(\alpha \right)$ be an extension of a number field $K$, where $\alpha$ satisfies the monic irreducible polynomial $P\left(X\right)=X^p-\beta$ of prime degree belonging to ${𝔬}_K\left[X\right]$ (${𝔬}_K$ is the ring of integers of $K$). The purpose of this paper is to study the monogenity of $L$ over $K$ by a simple and practical version of Dedekind’s criterion characterizing the existence of power integral bases over an arbitrary Dedekind ring by using the Gauss valuation and the index ideal. As an illustration, we determine an integral basis of a pure nonic field...

For any positive integer $k\ge 3$, it is easy to prove that the $k$-polygonal numbers are $a_n\left(k\right)=(2n+n(n-1)(k-2))/2$. The main purpose of this paper is, using the properties of Gauss sums and Dedekind sums, the mean square value theorem of Dirichlet $L$-functions and the analytic methods, to study the computational problem of one kind mean value of Dedekind sums $S(a_n\left(k\right){\overline{a}}_m\left(k\right),p)$ for $k$-polygonal numbers with $1\le m,n\le p-1$, and give an interesting computational formula for it.

General concepts and strategies are developed for identifying the isomorphism type of the second $p$-class group $G=\mathrm{Gal}\left({\mathrm{F}}_p^2\left(K\right)\right|K)$, that is the Galois group of the second Hilbert $p$-class field ${\mathrm{F}}_p^2\left(K\right)$, of a number field $K$, for a prime $p$. The isomorphism type determines the position of $G$ on one of the coclass graphs $𝒢(p,r)$, $r\ge 0$, in the sense of Eick, Leedham-Green, and Newman. It is shown that, for special types of the base field $K$ and of its $p$-class group ${\mathrm{Cl}}_p\left(K\right)$, the position of $G$ is restricted to certain admissible branches...

We give a simple proof that critical values of any Artin $L$-function attached to a representation $\rho$ with character ${\chi }_{\rho }$ are stable under twisting by a totally even character $\chi$, up to the $dim\rho$-th power of the Gauss sum related to $\chi$ and an element in the field generated by the values of ${\chi }_{\rho }$ and $\chi$ over $\mathbb{Q}$. This extends a result of Coates and Lichtenbaum as well as the previous work of Ward.

Let $\epsilon$ be an algebraic unit of the degree $n\ge 3$. Assume that the extension $\mathbb{Q}\left(\epsilon \right)/\mathbb{Q}$ is Galois. We would like to determine when the order $\mathbb{Z}\left[\epsilon \right]$ of $\mathbb{Q}\left(\epsilon \right)$ is $\mathrm{Gal}\left(\mathbb{Q}\right(\epsilon )/\mathbb{Q})$-invariant, i.e. when the $n$ complex conjugates ${\epsilon }_1,\cdots ,{\epsilon }_n$ of $\epsilon$ are in $\mathbb{Z}\left[\epsilon \right]$, which amounts to asking that $\mathbb{Z}[{\epsilon }_1,\cdots ,{\epsilon }_n]=\mathbb{Z}\left[\epsilon \right]$, i.e., that these two orders of $\mathbb{Q}\left(\epsilon \right)$ have the same discriminant. This problem has been solved only for $n=3$ by using an explicit formula for the discriminant of the order $\mathbb{Z}[{\epsilon }_1,{\epsilon }_2,{\epsilon }_3]$. However, there is no known similar formula for $n>3$. In the present paper, we put forward and...

For an algebraic number field $K$ with $3$-class group ${\mathrm{Cl}}_3\left(K\right)$ of type $(3,3)$, the structure of the $3$-class groups ${\mathrm{Cl}}_3\left(N_i\right)$ of the four unramified cyclic cubic extension fields $N_i$, $1\le i\le 4$, of $K$ is calculated with the aid of presentations for the metabelian Galois group ${\mathrm{G}}_3^2\left(K\right)=\mathrm{Gal}\left({\mathrm{F}}_3^2\left(K\right)\right|K)$ of the second Hilbert $3$-class field ${\mathrm{F}}_3^2\left(K\right)$ of $K$. In the case of a quadratic base field $K=\mathbb{Q}\left(\sqrt{D}\right)$ it is shown that the structure of the $3$-class groups of the four $S_3$-fields $N_1,...,N_4$ frequently determines the type of principalization of the $3$-class group of $K$ in $N_1,...,N_4$. This...

In $𝒟$-modules theory, Gauss-Manin systems are defined by the direct image of the structure sheaf $𝒪$ by a morphism. A major theorem says that these systems have only regular singularities. This paper examines the irregularity of an analogue of the Gauss-Manin systems. It consists in the direct image complex $f_{+}\left(𝒪{\mathrm{e}}^g\right)$ of a $𝒟$-module twisted by the exponential of a polynomial $g$ by another polynomial $f$, where $f$ and $g$ are two polynomials in two variables. The analogue of the Gauss-Manin systems can...

For any number field $K$ with non-elementary $3$-class group ${\mathrm{Cl}}_3\left(K\right)\simeq C_{3^e}\times C_3$, $e\ge 2$, the punctured capitulation type $\varkappa \left(K\right)$ of $K$ in its unramified cyclic cubic extensions $L_i$, $1\le i\le 4$, is an orbit under the action of $S_3\times S_3$. By means of Artin’s reciprocity law, the arithmetical invariant $\varkappa \left(K\right)$ is translated to the punctured transfer kernel type $\varkappa \left(G_2\right)$ of the automorphism group $G_2=\mathrm{Gal}({\mathrm{F}}_3^2\left(K\right)/K)$ of the second Hilbert $3$-class field of $K$. A classification of finite $3$-groups $G$ with low order and bicyclic commutator quotient $G/G^{\text{'}}\simeq C_{3^e}\times C_3$, $2\le e\le 6$, according to the algebraic...

Let p be a prime, ℤₚ be the finite field in p elements, k be a positive integer, and A be the multiplicative subgroup of nonzero kth powers in ℤₚ. The goal of this paper is to determine, for a given positive integer s, a value tₛ such that if |A| ≫ tₛ then every element of ℤₚ is a sum of s kth powers. We obtain $t₄=p^{22/39+ϵ}$, $t₅=p^{15/29+ϵ}$ and for s ≥ 6, $tₛ=p^{(9s+45)/(29s+33)+ϵ}$. For s ≥ 24 further improvements are made, such as $t_{32}=p^{5/16+ϵ}$ and $t_{128}=p^{1/4}$.