caa a22 4500 605496676 CHVBK 20210128100537.0 cr unu---uuuuu 210128e20150401xx s 000 0 eng 10.1007/s10231-013-0380-4 doi (NATIONALLICENCE)springer-10.1007/s10231-013-0380-4 On relations for rings generated by algebraic numbers and their conjugates [Elektronische Daten] [Paulius Drungilas, Artūras Dubickas, Jonas Jankauskas] Let $$\alpha $$ α be an algebraic number of degree $$d$$ d with minimal polynomial $$F \in \mathbb {Z}[X]$$ F ∈ Z [ X ] , and let $$\mathbb {Z}[\alpha ]$$ Z [ α ] be the ring generated by $$\alpha $$ α over $$\mathbb {Z}$$ Z . We are interested whether a given number $$\beta \in \mathbb {Q}(\alpha )$$ β ∈ Q ( α ) belongs to the ring $$\mathbb {Z}[\alpha ]$$ Z [ α ] or not. We give a practical computational algorithm to answer this question. Furthermore, we prove that a rational number $$r/t \in \mathbb {Q}$$ r / t ∈ Q , where $$r \in \mathbb {Z}, t \in \mathbb {N}, \gcd (r, t) = 1$$ r ∈ Z , t ∈ N , gcd ( r , t ) = 1 , belongs to the ring $$\mathbb {Z}[\alpha ]$$ Z [ α ] if and only if the square-free part of its denominator $$t$$ t divides all the coefficients of the minimal polynomial $$F \in \mathbb {Z}[X]$$ F ∈ Z [ X ] except for the constant coefficient $$F(0)$$ F ( 0 ) that must be relatively prime to $$t$$ t , namely $$\gcd (F(0),t)=1$$ gcd ( F ( 0 ) , t ) = 1 . We also study the question when the equality $$\mathbb {Z}[\alpha ] = \mathbb {Z}[\alpha ']$$ Z [ α ] = Z [ α ′ ] for algebraic numbers $$\alpha , \alpha '$$ α , α ′ conjugates over $$\mathbb {Q}$$ Q holds. In particular, it is shown that for each $$d \in \mathbb {N}$$ d ∈ N , there are conjugate algebraic numbers $$\alpha , \alpha '$$ α , α ′ of degree $$d$$ d satisfying $$\mathbb {Q}(\alpha ) = \mathbb {Q}(\alpha ')$$ Q ( α ) = Q ( α ′ ) and $$\mathbb {Z}[\alpha ] \ne \mathbb {Z}[\alpha ']$$ Z [ α ] ≠ Z [ α ′ ] . The question concerning the equality $$\mathbb {Z}[\alpha ]=\mathbb {Z}[\alpha ']$$ Z [ α ] = Z [ α ′ ] is answered completely for conjugate quadratic pairs $$\alpha ,\alpha '$$ α , α ′ and also for conjugate pairs $$\alpha , \alpha '$$ α , α ′ of cubic algebraic integers. Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg, 2013 Rings of algebraic numbers nationallicence Conjugate algebraic numbers nationallicence Minimal polynomial nationallicence Polynomials in finite fields nationallicence Algorithms nationallicence Drungilas Paulius Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225, Vilnius, Lithuania aut Dubickas Artūras Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225, Vilnius, Lithuania aut Jankauskas Jonas Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225, Vilnius, Lithuania aut Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/2(2015-04-01), 369-385 0373-3114 194:2<369 2015 194 10231 https://doi.org/10.1007/s10231-013-0380-4 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10231-013-0380-4 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Drungilas Paulius Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225, Vilnius, Lithuania aut NATIONALLICENCE 700 1- Dubickas Artūras Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225, Vilnius, Lithuania aut NATIONALLICENCE 700 1- Jankauskas Jonas Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225, Vilnius, Lithuania aut NATIONALLICENCE 773 0- Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/2(2015-04-01), 369-385 0373-3114 194:2<369 2015 194 10231