On relations for rings generated by algebraic numbers and their conjugates
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| 024 | 7 | 0 | |a 10.1007/s10231-013-0380-4 |2 doi |
| 035 | |a (NATIONALLICENCE)springer-10.1007/s10231-013-0380-4 | ||
| 245 | 0 | 0 | |a On relations for rings generated by algebraic numbers and their conjugates |h [Elektronische Daten] |c [Paulius Drungilas, Artūras Dubickas, Jonas Jankauskas] |
| 520 | 3 | |a 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. | |
| 540 | |a Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg, 2013 | ||
| 690 | 7 | |a Rings of algebraic numbers |2 nationallicence | |
| 690 | 7 | |a Conjugate algebraic numbers |2 nationallicence | |
| 690 | 7 | |a Minimal polynomial |2 nationallicence | |
| 690 | 7 | |a Polynomials in finite fields |2 nationallicence | |
| 690 | 7 | |a Algorithms |2 nationallicence | |
| 700 | 1 | |a Drungilas |D Paulius |u Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225, Vilnius, Lithuania |4 aut | |
| 700 | 1 | |a Dubickas |D Artūras |u Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225, Vilnius, Lithuania |4 aut | |
| 700 | 1 | |a Jankauskas |D Jonas |u Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225, Vilnius, Lithuania |4 aut | |
| 773 | 0 | |t Annali di Matematica Pura ed Applicata (1923 -) |d Springer Berlin Heidelberg |g 194/2(2015-04-01), 369-385 |x 0373-3114 |q 194:2<369 |1 2015 |2 194 |o 10231 | |
| 856 | 4 | 0 | |u https://doi.org/10.1007/s10231-013-0380-4 |q text/html |z Onlinezugriff via DOI |
| 898 | |a BK010053 |b XK010053 |c XK010000 | ||
| 900 | 7 | |a Metadata rights reserved |b Springer special CC-BY-NC licence |2 nationallicence | |
| 908 | |D 1 |a research-article |2 jats | ||
| 949 | |B NATIONALLICENCE |F NATIONALLICENCE |b NL-springer | ||
| 950 | |B NATIONALLICENCE |P 856 |E 40 |u https://doi.org/10.1007/s10231-013-0380-4 |q text/html |z Onlinezugriff via DOI | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Drungilas |D Paulius |u Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225, Vilnius, Lithuania |4 aut | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Dubickas |D Artūras |u Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225, Vilnius, Lithuania |4 aut | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Jankauskas |D Jonas |u Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225, Vilnius, Lithuania |4 aut | ||
| 950 | |B NATIONALLICENCE |P 773 |E 0- |t Annali di Matematica Pura ed Applicata (1923 -) |d Springer Berlin Heidelberg |g 194/2(2015-04-01), 369-385 |x 0373-3114 |q 194:2<369 |1 2015 |2 194 |o 10231 | ||