scholarly journals On a relative normal integral basis problem over abelian number fields

1993 ◽  
Vol 69 (10) ◽  
pp. 413-416 ◽  
Author(s):  
Humio Ichimura
Keyword(s):  
Number Fields ◽  
Integral Basis ◽  
Basis Problem ◽  
Normal 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

scholarly journals A normal integral basis Theorem

1980 ◽  
Vol 66 (2) ◽  
pp. 544-549 ◽  
Author(s):  
Alexander Sze
Keyword(s):  
Integral Basis ◽  
Basis Theorem ◽  
Normal Integral Basis

Module invariants and root numbers for quaternion fields of degree 41r

1974 ◽  
Vol 76 (2) ◽  
pp. 393-399 ◽  
Author(s):  
A. Fröhlich
Keyword(s):  
Module Structure ◽  
Integral Group ◽  
Galois Module ◽  
Integral Basis ◽  
Generators And Relations ◽  
Root Numbers ◽  
Normal Number ◽  
Image Position ◽  
Normal Integral Basis ◽  
Free Modules

1. The results. Let l be an odd prime, r ≥ 1, and letbe the quaternion group of order 4lr, as given by generators and relations. Throughout N is a tamely ramified normal number field with Galois group Gal (N/Q) = H (a ‘quaternion field’), and its ring of integers. We are interested in the structure of as a module over the integral group ring ZH. Deriving, first, certain classgroup invariants for locally free ZH-modules, we shall then determine those for the module in terms of the arithmetic invariants of N/Q. When 1 ≡ – 1 (mod 4), this yields again a Galois module interpretation of Artin root numbers quite analogous to that in (2). On the other hand for l ≡ 1 (mode 4), we shall get a weak ‘normal integral basis theorem’. The original impetus for this work came from computations of J. Queyrut, who – in different language – obtained these results in the case l = 3, r = 1 (cf. (7)). The tools, we are using, come from the general theory developed in recent years with such concrete applications in mind, and it is perhaps of interest to see how the various ‘strands’, on root numbers (cf. (3), (4)), on locally free modules (cf. (5)), and on Galois module structure (cf. (6)) are here pulled together. For technical reasons, we shall impose on N the slight further restriction, that l be non-ramified, although our results would remain true without this. Both the statements and the proofs of the theorem depend on ideas contained in (5) and (6). The reader who is prepared to take them for granted should, however, be able to read the present paper independently of those papers.

scholarly journals A normal integral basis theorem

1976 ◽  
Vol 39 (1) ◽  
pp. 131-137 ◽  
Author(s):  
A Fröhlich
Keyword(s):  
Integral Basis ◽  
Basis Theorem ◽  
Normal Integral Basis
Sign in / Sign up

Export Citation Format

Share Document