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.

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].

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.

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).

scholarly journals On Greenberg’s generalized conjecture for imaginary quartic fields

2020 ◽  
pp. 1-11
Author(s):  
Naoya Takahashi
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Number Field ◽  
Group Ring ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Prime Numbers ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Number Formula

For an algebraic number field [Formula: see text] and a prime number [Formula: see text], let [Formula: see text] be the maximal multiple [Formula: see text]-extension. Greenberg’s generalized conjecture (GGC) predicts that the Galois group of the maximal unramified abelian pro-[Formula: see text] extension of [Formula: see text] is pseudo-null over the completed group ring [Formula: see text]. We show that GGC holds for some imaginary quartic fields containing imaginary quadratic fields and some prime numbers.

scholarly journals On Block Idempotents of Modular Group Rings

1966 ◽  
Vol 27 (2) ◽  
pp. 429-433 ◽  
Author(s):  
Masaru Osima
Keyword(s):  
Prime Number ◽  
Finite Order ◽  
Number Field ◽  
Algebraic Number ◽  
Modular Group ◽  
Algebraic Number Field ◽  
Roots Of Unity ◽  
Group Rings ◽  
Irreducible Characters

We consider a group G of finite order g = pag′ where p is a prime number and (p, g′) = 1. Let Ω be the algebraic number field which contains the p-th roots of unity. Let K1, K2,…, Kn be the classes of conjugate elements in G and the first m(≦n) classes be p-regular. There exist n distinct (absolutely) irreducible characters x1, x2,…, xn of G.

scholarly journals Canonical ideals of Cohen-Macaulay partially ordered sets

1988 ◽  
Vol 112 ◽  
pp. 1-24 ◽  
Author(s):  
Takayuki Hibi
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Commutative Algebra ◽  
Partially Ordered Sets ◽  
Algebraic Number Field ◽  
Ideal Theory ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Ring Of Algebraic Integers ◽  
The Ideal

Our dream is to revive the ideal theory in partially ordered sets from a viewpoint of commutative algebra.Historically, the concept of ideals in commutative algebra was first studied by Dedekind, who considered the ring of algebraic integers in an algebraic number field.

On the Structure of Group Algebras, I

1965 ◽  
Vol 17 ◽  
pp. 583-593 ◽  
Author(s):  
James A. Cohn ◽  
Donald Livingstone
Keyword(s):  
Finite Group ◽  
Number Field ◽  
Algebraic Number ◽  
Special Class ◽  
Algebraic Number Field ◽  
Group Algebras ◽  
Basic Question ◽  
List Type ◽  
Ring Of Algebraic Integers ◽  
A Finite Group

With this paper we begin a study of the structure of the group algebra RG of a finite group G over the ring of algebraic integers R in an algebraic number field k. The basic question is whether non-isomorphic groups can have isomorphic algebras over R. We shall show that this is impossible if G is (a) abelian,(b) Hamiltonian,(c) one of a special class of p-groups.

On correspondence between orders and ideals with a normal basis in cyclic subfields of Q(ξp) of a prime degree l

2010 ◽  
Vol 60 (6) ◽  
Author(s):  
Juraj Kostra
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Normal Basis ◽  
Prime Degree ◽  
One To One

AbstractLet K be a tamely ramified cyclic algebraic number field of prime degree l. In the paper one-to-one correspondence between all orders of K with a normal basis and all ideals of K with a normal basis is given.

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