R-orders in a Split Algebra have Finitely Many Non-Isomorphic Irreducible Lattices as soon as R has Finite Class Number

1971 ◽  
Vol 14 (3) ◽  
pp. 405-409 ◽  
Author(s):  
Klaus W. Roggenkamp
Keyword(s):  
Finite Number ◽  
Number Field ◽  
Algebraic Number ◽  
Class Number ◽  
Algebraic Number Field ◽  
Dedekind Domain ◽  
Quotient Field ◽  
Ring Of Integers ◽  
Finite Dimensional

Let R be a Dedekind domain with quotient field K and ∧ an R-order in the finite-dimensional separable K-algebra A. If K is an algebraic number field with ring of integers R, then the Jordan-Zassenhaus theorem states that for every left A-module L, the set SL(M)={M: M=∧-lattice, KM≅L} splits into a finite number of nonisomorphic ∧-lattices (cf. Zassenhaus [5]).

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.

scholarly journals On some class number relations for Galois extensions

1987 ◽  
Vol 107 ◽  
pp. 121-133 ◽  
Author(s):  
Takashi Ono
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Class Number ◽  
Quadratic Forms ◽  
Algebraic Number Field ◽  
Narrow Sense ◽  
Binary Quadratic Forms ◽  
Finite Degree ◽  
Algebraic Tori ◽  
Number Of Classes

Let k be an algebraic number field of finite degree over Q, the field of rationals, and K be an extension of finite degree over k. By the use of the class number of algebraic tori, we can introduce an arithmetical invariant E(K/k) for the extension K/k. When k = Q and K is quadratic over Q, the formula of Gauss on the genera of binary quadratic forms, i.e. the formula where = the class number of K in the narrow sense, the number of classes is a genus of the norm form of K/Q and tK = the number of distinct prime factors of the discriminant of K, may be considered as an equality between E(K/Q) and other arithmetical invariants of K.

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]).

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.

Quadratic forms overR(t)

1966 ◽  
Vol 62 (2) ◽  
pp. 197-205 ◽  
Author(s):  
James T. Knight
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Class Number ◽  
Quadratic Forms ◽  
Rational Functions ◽  
Algebraic Number Field ◽  
Detailed Treatment ◽  
The Subject

AbstractA theory of quadratic forms is developed overR(t), the field of rational functions with real coefficients, instead of over an algebraic number field: some results are analogous to the classical ones, but many, in particular the non-finiteness of the class number, whose more detailed treatment will be the subject of a later paper, are not analogous.

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).’

scholarly journals Note on class number factors and prime decompositions

1977 ◽  
Vol 66 ◽  
pp. 167-182 ◽  
Author(s):  
Yoshiomi Furuta
Keyword(s):  
Algebraic Number ◽  
Class Number ◽  
Galois Extension ◽  
Class Group ◽  
Algebraic Number Field ◽  
Augmentation Ideal ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Finite Degree ◽  
Ring Of Integers

Let K be a Galois extension of an algebraic number field k of finite degree with Galois group g, be a congruent ideal class group of K, and M be the class field over K corresponding to . Assume that M is normal over k. Then g acts on as a group of automorphisms. Donote by lg the augmentation ideal of the group ring Zg over the ring of integers Z.

scholarly journals Bounds for the size of integral solutions to Ym = f(X)

1999 ◽  
Vol 42 (1) ◽  
pp. 127-141
Author(s):  
Dimitrios Poulakis
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Diophantine Equations ◽  
Algebraic Number Field ◽  
Upper Bounds ◽  
Ring Of Integers ◽  
Odd Order ◽  
Integral Solutions

Let K be an algebraic number field with ring of integers OK and f(X) ∈ OK[X]. In this paper we establish improved explicit upper bounds for the size of solutions in OK, of diophantine equations Y2 = f(X), where f(X) has at least three roots of odd order, and Ym = f(X), where m is an integer ≥ 3 and f(X) has at least two roots of order prime to m.

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.

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