A QUANTITATIVE ASPECT OF NON-UNIQUE FACTORIZATIONS: THE NARKIEWICZ CONSTANTS

2011 ◽  
Vol 07 (06) ◽  
pp. 1463-1502 ◽  
Author(s):  
WEIDONG GAO ◽  
ALFRED GEROLDINGER ◽  
QINGHONG WANG
Keyword(s):  
Number Theory ◽  
Algebraic Number ◽  
Class Group ◽  
Algebraic Number Field ◽  
Quantitative Aspect ◽  
Ring Of Integers ◽  
New Methods ◽  
Principal Ideals ◽  
Combinatorial Number Theory ◽  
Irreducible Elements

Let K be an algebraic number field with non-trivial class group G and let [Formula: see text] be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let Fk (x) denote the number of non-zero principal ideals [Formula: see text] with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that Fk (x) behaves, for x → ∞, asymptotically like x( log x)-1+1/|G|( log log x) N k(G). We study N k (G) with new methods from Combinatorial Number Theory.

A Stickelberger Condition on Cyclic Galois Extensions

1982 ◽  
Vol 34 (3) ◽  
pp. 686-690 ◽  
Author(s):  
L. N. Childs
Keyword(s):  
Abelian Group ◽  
Algebraic Number ◽  
Galois Extension ◽  
Class Group ◽  
Finite Abelian Group ◽  
Algebraic Number Field ◽  
Primitive Elements ◽  
Galois Extensions ◽  
Ring Of Integers ◽  
Isomorphism Classes

LetRbe a commutative ring,Ca finite abelian group,Sa Galois extension ofRwith groupC, in the sense of [1]. ViewingSas anRC-module defines the Picard invariant map [4] from the Harrison group Gal (R,C) of isomorphism classes of Galois extensions ofRwith groupCto CI (RC), the class group ofRC. The image of the Picard invariant map is known to be contained in the subgrouphCl (RC) of primitive elements of CI (RC) (for definition see below). Characterizing the image of the Picard invariant map has been of some interest, for the image describes the extent of failure of Galois extensions to have normal bases.LetRbe the ring of integers of an algebraic number fieldK.

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 The distribution of the irreducibles in an algebraic number field

2005 ◽  
Vol 79 (3) ◽  
pp. 369-390
Author(s):  
David M. Bradley ◽  
Ali E. Özlük ◽  
Rebecca A. Rozario ◽  
C. Snyder
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Principal Ideals ◽  
Irreducible Elements

AbstractWe study the distribution of principal ideals generated by irreducible elements in an algebraic number field.

scholarly journals A PRECISE RESULT ON THE ARITHMETIC OF NON-PRINCIPAL ORDERS IN ALGEBRAIC NUMBER FIELDS

2012 ◽  
Vol 11 (05) ◽  
pp. 1250087 ◽  
Author(s):  
ANDREAS PHILIPP
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Theoretical Approach ◽  
Class Group ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Precise Result ◽  
Principal Order

Let R be an order in an algebraic number field. If R is a principal order, then many explicit results on its arithmetic are available. Among others, R is half-factorial if and only if the class group of R has at most two elements. Much less is known for non-principal orders. Using a new semigroup theoretical approach, we study half-factoriality and further arithmetical properties for non-principal orders in algebraic number fields.

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 nilpotent factors of congruent ideal class groups of Galois extensions

1976 ◽  
Vol 62 ◽  
pp. 13-28 ◽  
Author(s):  
Yoshiomi Furuta
Keyword(s):  
Class Group ◽  
Algebraic Number Field ◽  
Augmentation Ideal ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Class Groups ◽  
Finite Degree ◽  
Ring Of Integers ◽  
Ideal Class Groups ◽  
The Ideal

Let K be a Galois extension of an algebraic number field k of finite degree with Galois group g. Then g acts on a congruent ideal class group of K as a group of automorphisms, when the class field M over K corresponding to is normal over K. Let Ig be the augmentation ideal of the group ring Zg over the ring of integers Z, namely Ig be the ideal of Zg generated by σ − 1, σ running over all elements of g. Then is the group of all elements aσ-1 where a and σ belong to and g respectively.

scholarly journals A generalization of Hubert’s Theorem 94

1984 ◽  
Vol 96 ◽  
pp. 83-94 ◽  
Author(s):  
Katsuya Miyake
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Class Group ◽  
Algebraic Number Field ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Finite Degree ◽  
Class Field ◽  
The Absolute

Let k be an algebraic number field of finite degree. We denote the absolute class field of k by and the absolute ideal class group of k by Cl(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]).

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