scholarly journals A generalization of simplest number fields and their integral basis

2020 ◽  
Author(s):  
L. Remete
Keyword(s):  
Number Fields ◽  
Integral Basis

scholarly journals A Handy Technique for Fundamental Unit in Specific Type of Real Quadratic Fields

2019 ◽  
Vol 5 (1) ◽  
pp. 495-498
Author(s):  
Özen Özer
Keyword(s):  
Number Theory ◽  
Class Number ◽  
Number Fields ◽  
Fundamental Unit ◽  
Quadratic Fields ◽  
Integral Basis ◽  
Real Quadratic Fields ◽  
Class Number Formula ◽  
Number Formula ◽  
Real Quadratic

AbstractDifferent types of number theories such as elementary number theory, algebraic number theory and computational number theory; algebra; cryptology; security and also other scientific fields like artificial intelligence use applications of quadratic fields. Quadratic fields can be separated into two parts such as imaginary quadratic fields and real quadratic fields. To work or determine the structure of real quadratic fields is more difficult than the imaginary one.The Dirichlet class number formula is defined as a special case of a more general class number formula satisfying any types of number field. It includes regulator, ℒ-function, Dedekind zeta function and discriminant for the field. The Dirichlet’s class number h(d) formula in real quadratic fields claims that we have h\left(d \right).log {\varepsilon _d} = \sqrt {\Delta} {\scr L} \left({1,\;{\chi _d}}\right) for positive d > 0 and the fundamental unit ɛd of {\rm{\mathbb Q}}\left({\sqrt d} \right) . It is seen that discriminant, ℒ-function and fundamental unit ɛd are significant and necessary tools for determining the structure of real quadratic fields.The focus of this paper is to determine structure of some special real quadratic fields for d > 0 and d ≡ 2,3 (mod4). In this paper, we provide a handy technique so as to calculate particular continued fraction expansion of integral basis element wd, fundamental unit ɛd, and so on for such real quadratic number fields. In this paper, we get fascinating results in the development of real quadratic fields.

scholarly journals ON INTEGRAL BASIS OF PURE NUMBER FIELDS

Mathematika ◽  
2020 ◽  
Vol 67 (1) ◽  
pp. 187-195
Author(s):  
Anuj Jakhar ◽  
Sudesh K. Khanduja ◽  
Neeraj Sangwan
Keyword(s):  
Number Fields ◽  
Integral Basis ◽  
Pure Number

Integral basis of pure prime degree number fields

2019 ◽  
Vol 50 (2) ◽  
pp. 309-314
Author(s):  
Anuj Jakhar ◽  
Neeraj Sangwan
Keyword(s):  
Number Fields ◽  
Integral Basis ◽  
Prime Degree

On power basis of a class of algebraic number fields

2016 ◽  
Vol 12 (08) ◽  
pp. 2317-2321 ◽  
Author(s):  
Bablesh Jhorar ◽  
Sudesh K. Khanduja
Keyword(s):  
Polynomial Ring ◽  
Algebraic Number ◽  
Minimal Polynomial ◽  
Sufficient Conditions ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Rational Numbers ◽  
Integral Basis ◽  
Algebraic Number Fields ◽  
Necessary And Sufficient

Let [Formula: see text] be an algebraic number field with [Formula: see text] in the ring [Formula: see text] of algebraic integers of [Formula: see text] and [Formula: see text] be the minimal polynomial of [Formula: see text] over the field [Formula: see text] of rational numbers. In 1977, Uchida proved that [Formula: see text] if and only if [Formula: see text] does not belong to [Formula: see text] for any maximal ideal [Formula: see text] of the polynomial ring [Formula: see text] (see [Osaka J. Math. 14 (1977) 155–157]). In this paper, we apply the above result to obtain some necessary and sufficient conditions involving the coefficients of [Formula: see text] for [Formula: see text] to equal [Formula: see text] when [Formula: see text] is a trinomial of the type [Formula: see text]. In the particular case when [Formula: see text], it is deduced that [Formula: see text] is an integral basis of [Formula: see text] if and only if either (i) [Formula: see text] and [Formula: see text] or (ii) [Formula: see text] divides [Formula: see text] and [Formula: see text].

scholarly journals Relative integral basis for algebraic number fields

1986 ◽  
Vol 9 (1) ◽  
pp. 97-104
Author(s):  
Mohmood Haghighi
Keyword(s):  
Algebraic Number ◽  
Number Fields ◽  
Integral Basis ◽  
Algebraic Number Fields ◽  
The Ideal

At first conditions are given for existence of a relative integral basis forOK≅Okn−1⊕Iwith[K;k]=n. Then the constrtiction of the idealIinOK≅Okn−1⊕Iis given for proof of existence of a relative integral basis forOK4(m1,m2)/Ok(​m3). Finally existence and construction of the relative integral basis forOK6(n3,−3)/Ok3(n3),OK6(n3,−3)/Ok2(−3)for some values ofnare given.

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.

scholarly journals Cutting towers of number fields

2021 ◽  
Author(s):  
Farshid Hajir ◽  
Christian Maire ◽  
Ravi Ramakrishna
Keyword(s):  
Number Fields
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