scholarly journals First-Degree Prime Ideals of Biquadratic Fields Dividing Prescribed Principal Ideals

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1433
Author(s):  
Giordano Santilli ◽  
Daniele Taufer
Keyword(s):  
Quadratic Fields ◽  
Prime Ideals ◽  
Prime Divisors ◽  
Principal Ideals ◽  
Biquadratic Fields

We describe first-degree prime ideals of biquadratic extensions in terms of the first-degree prime ideals of two underlying quadratic fields. The identification of the prime divisors is given by numerical conditions involving their ideal norms. The correspondence between these ideals in the larger ring and those in the smaller ones extends to the divisibility of specially-shaped principal ideals in their respective rings, with some exceptions that we explicitly characterize.

On the distribution of units modulo prime ideals in real quadratic fields

1998 ◽  
Vol 1998 (494) ◽  
pp. 65-72
Author(s):  
Masaru, Ishikawa ◽  
Yoshiyuki Kitaoka
Keyword(s):  
Quadratic Fields ◽  
Prime Ideals ◽  
Real Quadratic Fields ◽  
Real Quadratic

scholarly journals The distribution of prime ideals of imaginary quadratic fields

2003 ◽  
Vol 356 (2) ◽  
pp. 599-620
Author(s):  
G. Harman ◽  
A. Kumchev ◽  
P. A. Lewis
Keyword(s):  
Quadratic Fields ◽  
Prime Ideals ◽  
Imaginary Quadratic Fields

scholarly journals On asymptotic values of certain sets of attached prime ideals

1988 ◽  
Vol 30 (3) ◽  
pp. 293-300 ◽  
Author(s):  
A.-J. Taherizadeh
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Prime Ideals ◽  
Primary Decomposition ◽  
Prime Divisors ◽  
Finitely Generated Module ◽  
Commutative Noetherian Ring ◽  
Asymptotic Values ◽  
Attached Prime ◽  
Positive Integers

In his paper [1], M. Brodmann showed that if M is a1 finitely generated module over the commutative Noetherian ring R (with identity) and a is an ideal of R then the sequence of sets {Ass(M/anM)}n∈ℕ and {Ass(an−1M/anM)}n∈ℕ (where ℕ denotes the set of positive integers) are eventually constant. Since then, the theory of asymptotic prime divisors has been studied extensively: in [5], Chapters 1 and 2], for example, various results concerning the eventual stable values of Ass(R/an;) and Ass(an−1/an), denoted by A*(a) and B*(a) respectively, are discussed. It is worth mentioning that the above mentioned results of Brodmann still hold if one assumes only that A is a commutative ring (with identity) and M is a Noetherian A-module, and AssA(M), in this situation, is regarded as the set of prime ideals belonging to the zero submodule of M for primary decomposition.

scholarly journals ON THE -RANK OF CLASS GROUPS OF DIRICHLET BIQUADRATIC FIELDS

2020 ◽  
pp. 1-28
Author(s):  
Étienne Fouvry ◽  
Peter Koymans ◽  
Carlo Pagano
Keyword(s):  
Class Groups ◽  
Prime Divisors ◽  
Biquadratic Fields ◽  
Number Of Prime Divisors

Abstract We show that for $100\%$ of the odd, square free integers $n> 0$ , the $4$ -rank of $\text {Cl}(\mathbb{Q} (i, \sqrt {n}))$ is equal to $\omega _3(n) - 1$ , where $\omega _3$ is the number of prime divisors of n that are $3$ modulo $4$ .

Prime Ideals in GCD-Domains

1974 ◽  
Vol 26 (1) ◽  
pp. 98-107 ◽  
Author(s):  
Philip B. Sheldon
Keyword(s):  
General Setting ◽  
Chain Condition ◽  
Integral Closure ◽  
Prime Ideals ◽  
Unique Factorization Domain ◽  
Principal Ideals ◽  
Valuation Rings ◽  
Complete Integral Closure ◽  
Minimal Prime ◽  
Bezout Domains

A GCD-domain is a commutative integral domain in which each pair of elements has a greatest common divisor (g.c.d.). (This is the terminology of Kaplansky [9]. Bourbaki uses the term ''anneau pseudobezoutien" [3, p. 86], while Cohn refers to such rings as "HCF-rings" [4].) The concept of a GCD-domain provides a useful generalization of that of a unique factorization domain (UFD), since several of the standard results for a UFD can be proved in this more general setting (for example, integral closure, some properties of D[X], etc.). Since the class of GCD-domains contains all of the Bezout domains, and in particular, the valuation rings, it is clear that some of the properties of a UFD do not hold in general in a GCD-domain. Among these are complete integral closure, ascending chain condition on principal ideals, and some of the important properties of minimal prime ideals.

The Parity Distribution of Traces in Imaginary Quadratic Fields

1991 ◽  
Vol 34 (2) ◽  
pp. 196-201
Author(s):  
D. S. Dummit
Keyword(s):  
Class Group ◽  
Primary Component ◽  
Quadratic Fields ◽  
Prime Ideals ◽  
Imaginary Quadratic Fields ◽  
Parity Distribution

AbstractComputations of the Iwasawa λ -invariant for imaginary quadratic fields showed a discrepancy in the proportion of even and odd traces of certain integers from these imaginary quadratic fields. This paper shows that such a discrepancy is in some sense to be expected and that the proportion of even and odd traces of principal generators of powers of prime ideals in imaginary quadratic fields is related to the 3-primary component of the class group.

A note on prime divisors of polynomials P(Tk); k ≥ 1

2019 ◽  
Vol 69 (1) ◽  
pp. 213-222
Author(s):  
François Legrand
Keyword(s):  
Positive Integer ◽  
Number Field ◽  
Residue Field ◽  
Sufficient Conditions ◽  
Integral Closure ◽  
Prime Ideals ◽  
Prime Divisors ◽  
Reduction Modulo

Abstract Let F be a number field, OF the integral closure of ℤ in F, and P(T) ∈ OF[T] a monic separable polynomial such that P(0) ≠ 0 and P(1) ≠ 0. We give precise sufficient conditions on a given positive integer k for the following condition to hold: there exist infinitely many non-zero prime ideals 𝓟 of OF such that the reduction modulo 𝓟 of P(T) has a root in the residue field OF/𝓟, but the reduction modulo 𝓟 of P(Tk) has no root in OF/𝓟. This makes a result from a previous paper (motivated by a problem in field arithmetic) asserting that there exist (infinitely many) such integers k more precise.

LE 2-RANG DU GROUPE DE CLASSES DE CERTAINS CORPS BIQUADRATIQUES ET APPLICATIONS

2004 ◽  
Vol 15 (02) ◽  
pp. 169-182
Author(s):  
ABDELMALEK AZIZI ◽  
ALI MOUHIB
Keyword(s):  
Class Number ◽  
Class Group ◽  
Quadratic Field ◽  
Quadratic Fields ◽  
Class Field ◽  
Real Quadratic Fields ◽  
Biquadratic Fields ◽  
Class Field Tower ◽  
Real Quadratic

Let K be a real biquadratic field and let k be a quadratic field with odd class number contained in K. The aim of this article is to determine the rank of the 2-class group of K and we give applications to the structure of the 2-class group of some biquadratic fields and to the 2-class field tower of some real quadratic fields. Résumé: Soient K un corps biquadratique réel et k un sous-corps quadratique de K dont le nombre de classes est impair. Dans ce papier on détermine le rang du 2-groupe de classes de K et on donne des applications à la structure du 2-groupe de classes de certains corps biquadratiques et aussi à la tour des 2-corps de classes de Hilbert de certains corps quadratiques réels.

scholarly journals Imaginary bicyclic biquadratic fields with the real quadratic subfield of class-number one

1986 ◽  
Vol 102 ◽  
pp. 91-100
Author(s):  
Hideo Yokoi
Keyword(s):  
Class Number ◽  
The Other ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
The Real ◽  
Real Quadratic Fields ◽  
Other Hand ◽  
Biquadratic Fields ◽  
Real Quadratic

It has been proved by A. Baker [1] and H. M. Stark [7] that there exist exactly 9 imaginary quadratic fields of class-number one. On the other hand, G.F. Gauss has conjectured that there exist infinitely many real quadratic fields of class-number one, and the conjecture is now still unsolved.

scholarly journals Divisibility Criteria for Class Numbers of Imaginary Quadratic Fields Whose Discriminant Has Only Two Prime Factors

2012 ◽  
Vol 2012 ◽  
pp. 1-4
Author(s):  
A. Pekin
Keyword(s):  
Quadratic Fields ◽  
Class Numbers ◽  
Sufficient Condition ◽  
Prime Divisors ◽  
Imaginary Quadratic Fields ◽  
Prime Factors

We will prove a theorem providing sufficient condition for the divisibility of class numbers of certain imaginary quadratic fields by 2g, whereg>1is an integer and the discriminant of such fields has only two prime divisors.

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