On Gauss sums over dedekind domains

2021 ◽  
pp. 1-17
Author(s):  
Zhiyong Zheng ◽  
Man Chen ◽  
Jie Xu
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Integer ◽  
Algebraic Number Field ◽  
Gauss Sums ◽  
Algebraic Integers ◽  
Dedekind Domains ◽  
Finite Norm ◽  
Integer Ring ◽  
Ring Of Algebraic Integers

It is a difficult question to generalize Gauss sums to a ring of algebraic integers of an arbitrary algebraic number field. In this paper, we define and discuss Gauss sums over a Dedekind domain of finite norm. In particular, we give a Davenport–Hasse type formula for some special Gauss sums. As an application, we give some more precise formulas for Gauss sums over the algebraic integer ring of an algebraic number field (see Theorems 4.1 and 4.2).

Characterization of primes dividing the index of a trinomial

2017 ◽  
Vol 13 (10) ◽  
pp. 2505-2514 ◽  
Author(s):  
Anuj Jakhar ◽  
Sudesh K. Khanduja ◽  
Neeraj Sangwan
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Sufficient Conditions ◽  
Algebraic Number Field ◽  
Necessary And Sufficient Conditions ◽  
Algebraic Integers ◽  
Ring Of Algebraic Integers ◽  
Necessary And Sufficient

Let [Formula: see text] denote the ring of algebraic integers of an algebraic number field [Formula: see text], where [Formula: see text] is a root of an irreducible trinomial [Formula: see text] belonging to [Formula: see text]. In this paper, we give necessary and sufficient conditions involving only [Formula: see text] for a given prime [Formula: see text] to divide the index of the subgroup [Formula: see text] in [Formula: see text]. In particular, we deduce necessary and sufficient conditions for [Formula: see text] to be equal to [Formula: see text].

ON A THEOREM OF DEDEKIND

2008 ◽  
Vol 04 (06) ◽  
pp. 1019-1025 ◽  
Author(s):  
SUDESH K. KHANDUJA ◽  
MUNISH KUMAR
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Minimal Polynomial ◽  
Algebraic Number Field ◽  
Rational Numbers ◽  
Prime Ideals ◽  
Algebraic Integers ◽  
Dedekind Domains ◽  
Modulo P ◽  
Polynomial Formula

Let K = ℚ(θ) be an algebraic number field with θ in the ring AK of algebraic integers of K and f(x) be the minimal polynomial of θ over the field ℚ of rational numbers. For a rational prime p, let [Formula: see text] be the factorization of the polynomial [Formula: see text] obtained by replacing each coefficient of f(x) modulo p into product of powers of distinct monic irreducible polynomials over ℤ/pℤ. Dedekind proved that if p does not divide [AK : ℤ[θ]], then the factorization of pAK as a product of powers of distinct prime ideals is given by [Formula: see text], with 𝔭i = pAK + gi(θ)AK, and residual degree [Formula: see text]. In this paper, we prove that if the factorization of a rational prime p in AK satisfies the above-mentioned three properties, then p does not divide [AK:ℤ[θ]]. Indeed the analogue of the converse is proved for general Dedekind domains. The method of proof leads to a generalization of one more result of Dedekind which characterizes all rational primes p dividing the index of K.

Algebraic extensions in nonstandard models and Hilbert's irreducibility theorem

1988 ◽  
Vol 53 (2) ◽  
pp. 470-480 ◽  
Author(s):  
Masahiro Yasumoto
Keyword(s):  
Natural Number ◽  
Prime Number ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Algebraic Extension ◽  
Algebraic Integers ◽  
Nonstandard Models ◽  
Ring Of Algebraic Integers

LetKbe an algebraic number field andIKthe ring of algebraic integers inK. *Kand *IKdenote enlargements ofKandIKrespectively. LetxЄ *K–K. In this paper, we are concerned with algebraic extensions ofK(x)within *K. For eachxЄ *K–Kand each natural numberd, YK(x,d)is defined to be the number of algebraic extensions ofK(x)of degreedwithin *K.xЄ *K–Kis called a Hilbertian element ifYK(x,d)= 0 for alldЄ N,d> 1; in other words,K(x)has no algebraic extension within *K. In their paper [2], P. C. Gilmore and A. Robinson proved that the existence of a Hilbertian element is equivalent to Hilbert's irreducibility theorem. In a previous paper [9], we gave many Hilbertian elements of nonstandard integers explicitly, for example, for any nonstandard natural numberω, 2ωPωand 2ω(ω3+ 1) are Hilbertian elements in*Q, where pωis theωth prime number.

A Remark on Separable Orders

1969 ◽  
Vol 12 (4) ◽  
pp. 453-455 ◽  
Author(s):  
Klaus W. Roggenkamp
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Finitely Generated ◽  
Algebraic Integers ◽  
Finite Dimensional ◽  
Torsion Free

K = algebraic number field,R = algebraic integers in K,A = finite dimensional semi-simple K-algebra, A. = simple K-algebra,i = 1,…, n,Ki = center of Ai, = 1,…, n,G = R-order in A,Ri = G ∩ ki.All modules under consideration are finitely generated left modules. A G-lattice is a G-module which is R-torsion-free.

scholarly journals On some properties of group rings

1980 ◽  
Vol 29 (4) ◽  
pp. 385-392 ◽  
Author(s):  
G. Karpilovsky
Keyword(s):  
Commutative Ring ◽  
Number Field ◽  
Group Ring ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Group Rings ◽  
Arbitrary Group ◽  
Bijective Correspondence ◽  
Isomorphism Classes ◽  
Ring Of Algebraic Integers

AbstractLet Out (RG) be the set of all outer R-automorphisms of a group ring RG of arbitrary group G over a commutative ring R with 1. It is proved that there is a bijective correspondence between the set Out (RG) and a set consisting of R(G × G)-isomorphism classes of R-free R(G × G)-modules of a certain type. For the case when G is finite and R is the ring of algebraic integers of an algebraic number field the above result implies that there are only finitely many conjugacy classes of group bases in RG. A generalization of a result due to R. Sandling is also provided.

On the integral and globally irreducible representations of finite groups

2018 ◽  
Vol 17 (05) ◽  
pp. 1850087
Author(s):  
Dmitry Malinin
Keyword(s):  
Number Field ◽  
Finite Groups ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Integral Representations ◽  
Irreducible Representations ◽  
Minimal Realization ◽  
Algebraic Integers ◽  
Ring Of Integers ◽  
Representations Of Finite Groups

We consider the arithmetic of integral representations of finite groups over algebraic integers and the generalization of globally irreducible representations introduced by Van Oystaeyen and Zalesskii. For the ring of integers [Formula: see text] of an algebraic number field [Formula: see text] we are interested in the question: what are the conditions for subgroups [Formula: see text] such that [Formula: see text], the [Formula: see text]-span of [Formula: see text], coincides with [Formula: see text], the ring of [Formula: see text]-matrices over [Formula: see text], and what are the minimal realization fields.

scholarly journals Groups of Matrices With Integer Eigenvalues

1971 ◽  
Vol 12 (3) ◽  
pp. 351-357 ◽  
Author(s):  
M. R. Freislich
Keyword(s):  
Number Field ◽  
Characteristic Polynomial ◽  
Linear Group ◽  
Algebraic Number ◽  
General Linear Group ◽  
Algebraic Number Field ◽  
General Linear ◽  
Algebraic Numbers ◽  
Algebraic Integers

Let F be an algebraic number field, and S a subgroup of the general linear group GL(n, F). We shall call S a U-group if S satisfies the condition (U): Every x ∈ S is a matrix all of whose eigenvalues are algebraic integers. (This is equivalent to either of the following conditions: a) the eigenvalues of each matrix (x are all units as algebraic numbers; b) the characteristic polynomial for x has all its coefficients integers in F. In particular, then, every group of matrices with entries in the integers of F is a U-group.

scholarly journals On the approximation of algebraic numbers by algebraic integers

1963 ◽  
Vol 3 (4) ◽  
pp. 408-434 ◽  
Author(s):  
K. Mahler
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Number Field ◽  
Algebraic Number ◽  
Positive Constant ◽  
Algebraic Number Field ◽  
Algebraic Numbers ◽  
Algebraic Integers ◽  
Image Position ◽  
Rational Integral

In his Topics in Number Theory, vol. 2, chapter 2 (Reading, Mass., 1956) W. J. LeVeque proved an important generalisation of Roth's theorem (K. F. Roth, Mathematika 2,1955, 1—20).Let ξ be a fixed algebraic number, σ a positive constant, and K an algebraic number field of degree n. For κ∈K denote by κ(1), …, κ(n) the conjugates of κ relative to K, by h(κ) the smallest positive integer such that the polynomial has rational integral coefficients, and by q(κ) the quantity

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