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Power bases for rings of integers of abelian imaginary fields
Author(s) -
Ranieri Gabriele
Publication year - 2010
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdq032
Subject(s) - mathematics , ring of integers , abelian group , integer (computer science) , ring (chemistry) , discrete mathematics , algebraic number field , prime (order theory) , field (mathematics) , generator (circuit theory) , combinatorics , pure mathematics , power (physics) , physics , computer science , quantum mechanics , chemistry , organic chemistry , programming language
Let L be a number field and let L be its ring of integers. It is a very difficult problem to decide whether L has a power basis. Moreover, if a power basis exists, it is hard to find all the generators of L over ℤ. In this paper, we show that if α is a generator of the ring of integers of an abelian imaginary field whose conductor is relatively prime to 6, then either α is an integer translate of a root of unity, or α + α ¯is an odd integer. From this result and other remarks it follows that if β is a generator of the ring of integers of an abelian imaginary field with conductor relatively prime to 6 and β is not an integer translate of a root of unity, then β β ¯is a generator of the ring of integers of the maximal real field contained in ℚ( β ). Finally, we use a result of Gras to prove that if d > 1 is an integer relatively prime to 6, then all but finitely many imaginary extensions of ℚ of degree 2 d have a ring of integers that does not have a power basis.