NDLI: On relations for rings generated by algebraic numbers and their conjugates

Content Provider	Springer Nature Link
Author	Dubickas, Artūras Drungilas, Paulius Jankauskas, Jonas
Copyright Year	2013
Abstract	Let $$\alpha $$ be an algebraic number of degree $$d$$ with minimal polynomial $$F \in \mathbb {Z}[X]$$ , and let $$\mathbb {Z}[\alpha ]$$ be the ring generated by $$\alpha $$ over $$\mathbb {Z}$$ . We are interested whether a given number $$\beta \in \mathbb {Q}(\alpha )$$ belongs to the ring $$\mathbb {Z}[\alpha ]$$ or not. We give a practical computational algorithm to answer this question. Furthermore, we prove that a rational number $$r/t \in \mathbb {Q}$$ , where $$r \in \mathbb {Z}, t \in \mathbb {N}, \gcd (r, t) = 1$$ , belongs to the ring $$\mathbb {Z}[\alpha ]$$ if and only if the square-free part of its denominator $$t$$ divides all the coefficients of the minimal polynomial $$F \in \mathbb {Z}[X]$$ except for the constant coefficient $$F(0)$$ that must be relatively prime to $$t$$ , namely $$\gcd (F(0),t)=1$$ . We also study the question when the equality $$\mathbb {Z}[\alpha ] = \mathbb {Z}[\alpha ']$$ for algebraic numbers $$\alpha , \alpha '$$ conjugates over $$\mathbb {Q}$$ holds. In particular, it is shown that for each $$d \in \mathbb {N}$$ , there are conjugate algebraic numbers $$\alpha , \alpha '$$ of degree $$d$$ satisfying $$\mathbb {Q}(\alpha ) = \mathbb {Q}(\alpha ')$$ and $$\mathbb {Z}[\alpha ] \ne \mathbb {Z}[\alpha ']$$ . The question concerning the equality $$\mathbb {Z}[\alpha ]=\mathbb {Z}[\alpha ']$$ is answered completely for conjugate quadratic pairs $$\alpha ,\alpha '$$ and also for conjugate pairs $$\alpha , \alpha '$$ of cubic algebraic integers.
Ending Page	385
Page Count	17
Starting Page	369
File Format	PDF
ISSN	03733114
e-ISSN	16181891
Journal	Annali di Matematica Pura ed Applicata
Issue Number	2
Volume Number	194
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2013-10-29
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Polynomials (irreducibility, etc.) Rings of algebraic numbers Algorithms Polynomials Mathematics Minimal polynomial Conjugate algebraic numbers Polynomials in finite fields Algorithms; complexity Algebraic numbers; rings of algebraic integers
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

Several types of algebraic numbers on the unit circle

A generalization of simplest number fields and their integral basis

The Weil height in terms of an auxiliary polynomial

Minimal Degrees of Algebraic Numbers with respect to Primitive Elements

The t-metric Mahler measures of surds and rational numbers

Finite polynomial orbits in finitely generated domains

Two Families of Monogenic $S_4$ Quartic Number Fields

Julia Robinson numbers and arithmetical dynamic of quadratic polynomials

The $t$-metric Mahler measures of surds and rational numbers

On relations for rings generated by algebraic numbers and their conjugates

Similar Documents

Several types of algebraic numbers on the unit circle

A generalization of simplest number fields and their integral basis

The Weil height in terms of an auxiliary polynomial

Minimal Degrees of Algebraic Numbers with respect to Primitive Elements

The t-metric Mahler measures of surds and rational numbers

Finite polynomial orbits in finitely generated domains

Two Families of Monogenic $S_4$ Quartic Number Fields

Julia Robinson numbers and arithmetical dynamic of quadratic polynomials

The $t$-metric Mahler measures of surds and rational numbers

On relations for rings generated by algebraic numbers and their conjugates