On relations for rings generated by algebraic numbers and their conjugates

• Published in: Annali di matematica pura ed applicata 2015-04, Vol.194 (2), p.369-385
• Main Authors: Drungilas, Paulius, Dubickas, Artūras, Jankauskas, Jonas
• Format: Article
• Language: English
• Subjects: Algebra; Algorithms; Conjugates; Mathematical analysis; Mathematics; Mathematics and Statistics; Polynomials; Questions; Rings (mathematics)
• ISSN: 0373-3114; 1618-1891
• DOI: 10.1007/s10231-013-0380-4

Let α be an algebraic number of degree d with minimal polynomial F ∈ Z [ X ] , and let Z [ α ] be the ring generated by α over Z . We are interested whether a given number β ∈ Q ( α ) belongs to the ring Z [ α ] or not. We give a practical computational algorithm to answer this question. Furthermore, we prove that a rational number r / t ∈ Q , where r ∈ Z , t ∈ N , gcd ( r , t ) = 1 , belongs to the ring Z [ α ] if and only if the square-free part of its denominator t divides all the coefficients of the minimal polynomial F ∈ 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 Z [ α ] = Z [ α ′ ] for algebraic numbers α , α ′ conjugates over Q holds. In particular, it is shown that for each d ∈ N , there are conjugate algebraic numbers α , α ′ of degree d satisfying Q ( α ) = Q ( α ′ ) and   Z [ α ] ≠ Z [ α ′ ] . The question concerning the equality Z [ α ] = Z [ α ′ ] is answered completely for conjugate quadratic pairs α , α ′ and also for conjugate pairs α , α ′ of cubic algebraic integers.