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 On the Ring of Integers in an Algebraic Number Field as a representation Module of Galois Group

1960 ◽  
Vol 16 ◽  
pp. 83-90 ◽  
Author(s):  
Hideo Yokoi
Keyword(s):  
Irreducible Representation ◽  
Galois Group ◽  
Number Field ◽  
Algebraic Number ◽  
Prime Order ◽  
Algebraic Number Field ◽  
Integral Representations ◽  
Indecomposable Representation ◽  
List Type ◽  
Ring Of Integers

1. Introduction. It is known that there are only three rationally inequivalent classes of indecomposable integral representations of a cyclic group of prime order l. The representations of these classes are: (I) identical representation,(II) rationally irreducible representation of degree l – 1,(III) indecomposable representation consisting of one identical representation and one rationally irreducible representation of degree l-1 (F. E. Diederichsen [1], I. Reiner [2]).

On integral absolutely irreducible representations of finite groups

2016 ◽  
Vol 15 (03) ◽  
pp. 1650048 ◽  
Author(s):  
Dmitry Malinin
Keyword(s):  
Infinite Series ◽  
Finite Groups ◽  
Algebraic Number ◽  
Integral Representations ◽  
Irreducible Representations ◽  
Two Dimensional ◽  
Representations Of Finite Groups

We consider the arithmetic background of integral representations of finite groups over [Formula: see text]-adic and algebraic number rings. Some infinite series of integral pairwise inequivalent absolutely irreducible representations of finite [Formula: see text]-groups with the extra congruence conditions are constructed, and some applications are given. Certain problems concerning integral irreducible two-dimensional representations over number rings are discussed.

scholarly journals On the isomorphism class of the ring of all integers of a cyclic wildly ramified extension of degree p II

1988 ◽  
Vol 111 ◽  
pp. 165-171 ◽  
Author(s):  
Yoshimasa Miyata
Keyword(s):  
Number Field ◽  
Cyclic Group ◽  
Algebraic Number ◽  
Isomorphism Class ◽  
Algebraic Number Field ◽  
Ring Of Integers

Let k be an algebraic number field with the ring of integers ok = o and let G be a cyclic group of order p, an odd prime.

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.

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 problem in algebraic number theory

1996 ◽  
Vol 119 (2) ◽  
pp. 191-200 ◽  
Author(s):  
J. Wójcik
Keyword(s):  
Number Theory ◽  
Prime Ideal ◽  
Number Field ◽  
Algebraic Number ◽  
Multiplicative Group ◽  
Algebraic Number Field ◽  
Algebraic Number Theory ◽  
Ring Of Integers

Let K be an algebraic number field. If q is a prime ideal of the ring of integers of K and α is a number of K prime to q then Mq(α) denotes the multiplicative group generated by α modulo q. In the paper [5] there is the remark: ‘We do not know whether for all a, b, c ∈ ℚ with abc ≠ 0, |a| ≠ 1,|b| ≠ 1,|c| ≠ 1 there exist infinitely many primes q with Mq (a) = Mq (b) = Mq (c).’

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.

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