Let $(K,ν)$ be an arbitrary-rank valued field, $R_ν$ its valuation ring, $K(α)/K$ a separable finite field extension generated over $K$ by a root of a monic irreducible polynomial $f\in R_ν[X]$. We give necessary and sufficient conditions for $R_ν[α]$ to be integrally closed. We further characterize the integral closedness of $R_ν[α]$ based on information about the valuations on $K(α)$ extending $ν$. Our results enhance and generalize some existing results in the relevant literature. Some applications and examples are also given.
In this article we show how the Dedekind-Hasse criterion may be applied to prove a simple result about quadratic number fields that usually is derived as a consequence of the theory of ideals and ideal classes.
We compare three different characterizations, due respectively to R. Dedekind, K. Uchida, and H. Lüneburg, of when $\mathbb Z[θ]$ is the ring of integers of $\mathbb Q(θ)$, and apply our results to some concrete $2$-towers of number fields.
In [3] it was shown that the Dedekond sums $s(m_1,n)$ and $s(m_2,n)$ are equal only if $(m_1m_2-1)(m_1-m_2)\equiv 0$ mod $n$. Here we show that the latter condition is equivalent to $12s(m_1,n)-12s(m_2,n)\in \Z$. In addition, we determine, for a given number $m_1$, the number of integers $m_2$ in the range $0\le m_2<n$, $(m_1,m_2)=1$, such that $12s(m_1,n)-12s(m_2,n)\in \Z$, provided that $n$ is square-free.
We give a number field analogue of a result of Ramanujan, Hardy and Littlewood, thereby obtaining a modular relation involving the non-trivial zeros of the Dedekind zeta function. We also provide a Riesz-type criterion for the Generalized Riemann Hypothesis for $ζ_K(s)$. New elegant transformations are obtained when $K$ is a quadratic extension, one of which involves the modified Bessel function of the second kind.
Let $R$ be a Dedekind ring, $K$ its quotient field, and $L=K(α)$ a finite field extension of $K$ defined by a monic irreducible polynomial $f(x)\in R[x]$. We give an easy version of Dedekind's criterion which computationally improves those versions know in the literature. We further use this result to give a sufficient condition for the integral closedness of $R[α]$ when $f(x)=x^n-a$. In case $R$ is a ring of integers of a number field, we give yet sufficient and necessary conditions for this to hold, generalizing and improving in both cases some known results in this direction. Some highlight…
Two number fields are called arithmetically equivalent if they have the same Dedekind zeta function. In the 1970's Perlis showed that this is equivalent to the condition that for almost every rational prime $\ell$ the arithmetic type of $\ell$ is the same in each field. In the 1990's Perlis and Stuart gave an unexpected characterization for arithmetic equivalence; they showed that to be arithmetically equivalent it is enough for almost every prime $\ell$ to have the same number of prime factors in each field. Here, using an $\ell-p$ switch trick, we provide an elementary proof of that fact bas…
For an Anderson A-motive over a discretely valued field whose residue field has A-characteristic ε, we prove a criterion for good reduction in terms of its associated local shtuka at ε. This yields a criterion for good reduction of Drinfeld modules. Our criterion is the function-field analog of Grothendieck's and de Jong's criterion for good reduction of an abelian variety over a discretely valued field with residue characteristic p in terms of its associated p-divisible group.
We address the question of non-vanishing of $L(1,f)$ where $f$ is an algebraic-valued, periodic arithmetical function. We do this by characterizing algebraic-valued, periodic functions $f$ for which $L(1,f)=0$. The case of odd functions was resolved by Baker, Birch and Wirsing in 1973. We apply a result of Bass to obtain a characterization for the even functions. We also describe a theorem of the first two authors which says that it is enough to consider only the even and the odd functions in order to obtain a complete characterization.
We give a simple characterization of the totally wild ramified valuations in a Galois extension of fields of characteristic p. This criterion involves the valuations of Artin-Schreier cosets of the F_{p^r}^\times-translation of a single element. We apply the criterion to construct some interesting examples.
Vector-Valued Period Polynomials and Zeta Values of Quadratic Fields
Institute of Basic Science, Korea University（韩国科学技术院基础科学研究所，韩国大学）；Department of Mathematics Education, Sungkyunkwan University（成均馆大学数学教育系）；American University of Beirut (AUB)（贝鲁特美国大学）
We consider a family S=S(a) of 2-valued transformations of special form on the segment [0,1] with measure $μ=\int p(x) dλ$, which is absolutely continuous with respect to the Lebesgue measure $λ$. We endow S with a set of weight functions $α=\{α_1(x),α_2(x)\}$ and find a criterion of measure invariance under the transformation. This criterion relates the three parameters $a$, $p$, $α$ to each other.
For an integral domain $R$, the {\it ring of integer-valued polynomials} over $R$ consists of all polynomials $f(X) \in R[X]$ such that $f(R) \subseteq R$. An interesting case to study is when $R$ is a Dedekind domain, in particular when $R$ is the ring of integers of an algebraic number field. An algebraic number field $K$ with ring of integers $\mathcal{O}_{K}$ is said to be a Pólya field if the $\mathcal{O}_{K}$-module of integer-valued polynomials on $K$ admits a regular basis. Associated to $K$ is a subgroup $Po(K)$ of the ideal class group $Cl_{K}$, known as the {\it Pólya group of $K$},…
Let $K$ be a valued field (in general $K$ is not heselian) with valuation $v$ and $A(x)\in K[x]$ be a polynomial of degree $n$. We find necessary and sufficient conditions for the existence of the elements $s,t,u\in K$, $s\neq 0\neq u$, such that at least one of the polynomials $u^nA(\frac{sx+t}{u})$, $(tx+u)^nA(\frac{sx}{tx+u})$, $(ux)^nA(\frac{tx+s}{ux})$ or $(ux+t)^nA(\frac{s}{ux+t})$ is an Eisenstein-Dumas polynomial at $v$, provided that the characteristic of the residue field of $v$ does not divide $n$. Furthermore, we show that if the orbit $A(x)GL(2,K)$ contains an Eisenstein-Dumas pol…
In this article, we prove an extension property of semipositively metrized ample invertible sheaves on a projective scheme over a complete non-archimedean valued field. As an application, we establish a Nakai-Moishezon type criterion for adelically normed graded linear series.
Let $K$ be a number field of degree $n$ with ring of integers $O_K$. By means of a criterion of Gilmer for polynomially dense subsets of the ring of integers of a number field, we show that, if $h\in K[X]$ maps every element of $O_K$ of degree $n$ to an algebraic integer, then $h(X)$ is integral-valued over $O_K$, that is $h(O_K)\subset O_K$. A similar property holds if we consider the set of all algebraic integers of degree $n$ and a polynomial $f\in\mathbb{Q}[X]$: if $f(α)$ is integral over $\mathbb{Z}$ for every algebraic integer $α$ of degree $n$, then $f(β)$ is integral over $\mathbb{Z}$ …
