Indices in a number field II

2019 ◽  
Vol 15 (01) ◽  
pp. 89-103
Author(s):  
Mohamed Ayad ◽  
Rachid Bouchenna ◽  
Omar Kihel
Keyword(s):  
Prime Number ◽  
Number Field ◽  
Additive Group ◽  
Necessary Condition ◽  
Ring Of Integers ◽  
Number Formula

Let [Formula: see text] be a number field of degree [Formula: see text] over [Formula: see text] and [Formula: see text] its ring of integers. For a prime number [Formula: see text], we determine the types of splittings of [Formula: see text] in [Formula: see text] for which the set [Formula: see text] is of cardinality a power of [Formula: see text]. We prove that this necessary condition is also sufficient for [Formula: see text] to be a subgroup of the additive group [Formula: see text]. Consequently, we show that, in this case, the subset of [Formula: see text], [Formula: see text] is an order of the number field.

scholarly journals The lattice of primary ideals of orders in quadratic number fields

2016 ◽  
Vol 12 (07) ◽  
pp. 2025-2040 ◽  
Author(s):  
Giulio Peruginelli ◽  
Paolo Zanardo
Keyword(s):  
Prime Ideal ◽  
Prime Number ◽  
Number Field ◽  
Number Fields ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Ring Of Integers ◽  
The Core ◽  
Number Formula ◽  
Primary Ideals

Let [Formula: see text] be an order in a quadratic number field [Formula: see text] with ring of integers [Formula: see text], such that the conductor [Formula: see text] is a prime ideal of [Formula: see text], where [Formula: see text] is a prime. We give a complete description of the [Formula: see text]-primary ideals of [Formula: see text]. They form a lattice with a particular structure by layers; the first layer, which is the core of the lattice, consists of those [Formula: see text]-primary ideals not contained in [Formula: see text]. We get three different cases, according to whether the prime number [Formula: see text] is split, inert or ramified in [Formula: see text].

Reductions of one-dimensional tori

2016 ◽  
Vol 13 (06) ◽  
pp. 1473-1489 ◽  
Author(s):  
Antonella Perucca
Keyword(s):  
Prime Number ◽  
Number Field ◽  
Rational Points ◽  
Splitting Field ◽  
Dimensional Torus ◽  
Finitely Generated ◽  
One Dimensional ◽  
Finitely Generated Group ◽  
Dirichlet Density ◽  
Number Formula

Consider a non-split one-dimensional torus defined over a number field [Formula: see text]. For a finitely generated group [Formula: see text] of rational points and for a prime number [Formula: see text], we investigate for how many primes [Formula: see text] of [Formula: see text] the size of the reduction of [Formula: see text] modulo [Formula: see text] is coprime to [Formula: see text]. We provide closed formulas for the corresponding Dirichlet density in terms of finitely many computable parameters. To achieve this, we determine in general which torsion fields and Kummer extensions contain the splitting field.

Iwasawa Theory for K2n

2013 ◽  
Vol 12 (1) ◽  
pp. 115-123
Author(s):  
Qingzhong Ji ◽  
Hourong Qin
Keyword(s):  
Prime Number ◽  
Number Field ◽  
Iwasawa Theory ◽  
Ring Of Integers ◽  
Cyclotomic Extension

AbstractGiven a number field F and a prime number p; let Fn denote the cyclotomic extension with [Fn : F] = pn; and let $\mathematical script capital(O)_F_n\$ denote its ring of integers. We establish an analogue of the classical Iwasawa theorem for the orders of K2i ($\mathematical script capital(O)_F_n\$){p}.

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.

FINITENESS RESULTS FOR REGULAR DEFINITE TERNARY QUADRATIC FORMS OVER ${\mathbb Q}(\sqrt{5})$

2007 ◽  
Vol 03 (04) ◽  
pp. 541-556 ◽  
Author(s):  
WAI KIU CHAN ◽  
A. G. EARNEST ◽  
MARIA INES ICAZA ◽  
JI YOUNG KIM
Keyword(s):  
Quadratic Form ◽  
Number Field ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Integral Quadratic Form ◽  
Ring Of Integers ◽  
Ternary Quadratic Forms ◽  
Totally Positive ◽  
Finiteness Result

Let 𝔬 be the ring of integers in a number field. An integral quadratic form over 𝔬 is called regular if it represents all integers in 𝔬 that are represented by its genus. In [13,14] Watson proved that there are only finitely many inequivalent positive definite primitive integral regular ternary quadratic forms over ℤ. In this paper, we generalize Watson's result to totally positive regular ternary quadratic forms over [Formula: see text]. We also show that the same finiteness result holds for totally positive definite spinor regular ternary quadratic forms over [Formula: see text], and thus extends the corresponding finiteness results for spinor regular quadratic forms over ℤ obtained in [1,3].

IWASAWA TYPE FORMULA FOR COVERS OF A LINK IN A RATIONAL HOMOLOGY SPHERE

2008 ◽  
Vol 17 (10) ◽  
pp. 1199-1221 ◽  
Author(s):  
TERUHISA KADOKAMI ◽  
YASUSHI MIZUSAWA
Keyword(s):  
Number Field ◽  
Class Number ◽  
Homology Sphere ◽  
Rational Homology ◽  
Homology Groups ◽  
Cyclic Covers ◽  
Class Number Formula ◽  
Number Formula ◽  
Iwasawa Invariants ◽  
Rational Homology Sphere

Based on the analogy between links and primes, we present an analogue of the Iwasawa's class number formula in a Zp-extension for the p-homology groups of pn-fold cyclic covers of a link in a rational homology 3-sphere. We also describe the associated Iwasawa invariants precisely for some examples and discuss analogies with the number field case.

On a Theorem of Kawamoto on Normal Bases of Rings of Integers, II

2005 ◽  
Vol 48 (4) ◽  
pp. 576-579 ◽  
Author(s):  
Humio Ichimura
Keyword(s):  
Prime Number ◽  
Number Field ◽  
Class Group ◽  
Natural Homomorphism ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Integral Basis ◽  
Rings Of Integers ◽  
Integer Ring ◽  
The Ideal

AbstractLet m = pe be a power of a prime number p. We say that a number field F satisfies the property when for any a ∈ F×, the cyclic extension F(ζm, a1/m)/F(ζm) has a normal p-integral basis. We prove that F satisfies if and only if the natural homomorphism is trivial. Here K = F(ζm), and denotes the ideal class group of F with respect to the p-integer ring of F.

scholarly journals p-Torsion points on elliptic curves defined over quadratic fields

1984 ◽  
Vol 96 ◽  
pp. 139-165 ◽  
Author(s):  
Fumiyuki Momose
Keyword(s):  
Elliptic Curves ◽  
Prime Number ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Quadratic Fields ◽  
Finite Degree ◽  
Torsion Points

Let p be a prime number and k an algebraic number field of finite degree d. Manin [14] showed that there exists an integer n = n(k,p) (≧0) which satisfies the condition

scholarly journals Galois Group of the Maximal Abelian Extension over an Algebraic Number Field

1957 ◽  
Vol 12 ◽  
pp. 177-189 ◽  
Author(s):  
Tomio Kubota
Keyword(s):  
Field Theory ◽  
Galois Group ◽  
Prime Number ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Abelian Extension ◽  
Finite Degree ◽  
Class Field ◽  
Continuous Character

The aim of the present work is to determine the Galois group of the maximal abelian extension ΩA over an algebraic number field Ω of finite degree, which we fix once for all.Let Z be a continuous character of the Galois group of ΩA/Ω. Then, by class field theory, the character Z is also regarded as a character of the idele group of Ω. We call such Z character of Ω. For our purpose, it suffices to determine the group Xl of the characters of Ω whose orders are powers of a prime number l.

D(n)-quadruples in the ring of integers of ℚ(√2, √3)

2019 ◽  
Vol 69 (6) ◽  
pp. 1263-1278
Author(s):  
Zrinka Franušić ◽  
Borka Jadrijević
Keyword(s):  
Number Field ◽  
Ring Of Integers

Abstract Let 𝓞𝕂 be the ring of integers of the number field 𝕂 = $\begin{array}{} \displaystyle \mathbb{Q}(\sqrt{2},\sqrt{3}) \end{array}$. A D(n)-quadruple in the ring 𝓞𝕂 is a set of four distinct non-zero elements {z1, z2, z3, z4} ⊂ 𝓞𝕂 with the property that the product of each two distinct elements increased by n is a perfect square in 𝓞𝕂. We show that the set of all n ∈ 𝓞𝕂 such that a D(n)-quadruple in 𝓞𝕂 exists coincides with the set of all integers in 𝕂 that can be represented as a difference of two squares of integers in 𝕂.

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