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An extension of the theorem on primitive divisors in algebraic number fields


Author: A. Schinzel
Journal: Math. Comp. 61 (1993), 441-444
MSC: Primary 11R47; Secondary 11R04
DOI: https://doi.org/10.1090/S0025-5718-1993-1189523-2
MathSciNet review: 1189523
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Abstract: The theorem about primitive divisors in algebraic number fields is generalized in the following manner. Let A, B be algebraic integers, $ (A,B) = 1, AB \ne 0$, A/B not a root of unity, and $ {\zeta _k}$ a primitive root of unity of order k. For all sufficiently large n, the number $ {A^n} - {\zeta _k}{B^n}$ has a prime ideal factor that does not divide $ {A^m} - \zeta _k^j{B^m}$ for arbitrary $ m < n$ and $ j < k$.


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DOI: https://doi.org/10.1090/S0025-5718-1993-1189523-2
Article copyright: © Copyright 1993 American Mathematical Society