scholarly journals Higher Newton polygons in the computation of discriminants and prime ideal decomposition in number fields

2011 ◽  
Vol 23 (3) ◽  
pp. 667-696 ◽  
Author(s):  
Jordi Guàrdia ◽  
Jesús Montes ◽  
Enric Nart
Keyword(s):  
Prime Ideal ◽  
Number Fields ◽  
Newton Polygons ◽  
Ideal Decomposition

scholarly journals The least unramified prime which does not split completely

2018 ◽  
Vol 30 (3) ◽  
pp. 651-661 ◽  
Author(s):  
Asif Zaman
Keyword(s):  
Prime Ideal ◽  
Finite Extension ◽  
Number Fields ◽  
Upper Bounds ◽  
Effective Field

AbstractLet {K/F} be a finite extension of number fields of degree {n\geq 2}. We establish effective field-uniform unconditional upper bounds for the least norm of a prime ideal {\mathfrak{p}} of F which is degree 1 over {\mathbb{Q}} and does not ramify or split completely in K. We improve upon the previous best known general estimates due to Li [7] when {F=\mathbb{Q}}, and Murty and Patankar [9] when {K/F} is Galois. Our bounds are the first when {K/F} is not assumed to be Galois and {F\neq\mathbb{Q}}.

scholarly journals Distribution of units of real quadratic number fields

2000 ◽  
Vol 158 ◽  
pp. 167-184 ◽  
Author(s):  
Yen-Mei J. Chen ◽  
Yoshiyuki Kitaoka ◽  
Jing Yu
Keyword(s):  
Prime Ideal ◽  
Quadratic Field ◽  
Number Fields ◽  
Prime Ideals ◽  
Positive Density ◽  
Real Quadratic Field ◽  
Quadratic Number ◽  
Ring Of Integers ◽  
Group Of Units ◽  
Real Quadratic

AbstractLet k be a real quadratic field and k, E the ring of integers and the group of units in k. Denoting by E() the subgroup represented by E of (k/)× for a prime ideal , we show that prime ideals for which the order of E() is theoretically maximal have a positive density under the Generalized Riemann Hypothesis.

scholarly journals Index, the prime ideal factorization in simplest quartic fields and counting their discriminants

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 591-600
Author(s):  
Abdelmejid Bayad ◽  
Mohammed Seddik
Keyword(s):  
Asymptotic Formula ◽  
Prime Ideal ◽  
Number Fields ◽  
The Other ◽  
Other Hand ◽  
Quartic Polynomials

We consider the simplest quartic number fields Km defined by the irreducible quartic polynomials x4-mx3-6x2+mx+1, where m runs over the positive rational integers such that the odd part of m2+16 is square free. In this paper, we study the index I(Km) and determine the explicit prime ideal factorization of rational primes in simplest quartic number fields Km. On the other hand, we establish an asymptotic formula for the number of simplest quartic fields with discriminant ? x and given index.

Prime ideal decomposition in $$F({}^m\sqrt \mu )$$

1976 ◽  
Vol 81 (2) ◽  
pp. 131-139 ◽  
Author(s):  
Henry B. Mann ◽  
William Yslas V�lez
Keyword(s):  
Prime Ideal ◽  
Ideal Decomposition

scholarly journals Prime ideal decomposition inF(μ1∕p)

1978 ◽  
Vol 75 (2) ◽  
pp. 589-600 ◽  
Author(s):  
William Vélez
Keyword(s):  
Prime Ideal ◽  
Ideal Decomposition

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].

NEWTON POLYGONS AND p-INTEGRAL BASES OF QUARTIC NUMBER FIELDS

2012 ◽  
Vol 11 (04) ◽  
pp. 1250073 ◽  
Author(s):  
LHOUSSAIN EL FADIL ◽  
JESÚS MONTES ◽  
ENRIC NART
Keyword(s):  
Prime Number ◽  
Number Fields ◽  
Explicit Description ◽  
Newton Polygons ◽  
Integral Basis ◽  
Integral Bases ◽  
Quartic Field ◽  
Potential Applications

Let p be a prime number. In this paper we use an old technique of Ø. Ore, based on Newton polygons, to construct in an efficient way p-integral bases of number fields defined by p-regular equations. To illustrate the potential applications of this construction, we derive from this result an explicit description of a p-integral basis of an arbitrary quartic field in terms of a defining equation.

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