Abstract
Let K be an algebraic extension field of Q. Further, let 𝔒K be the domain of algebraic integers of K, and let Σn(x) be the n-th cyclotomic polynomial. This paper is devoted to the factorization of the principal ideal Σn(e) 𝔒K(e𝔒K) into prime ideals of 𝔒K. The main result (Theorem 3.4) can be considered as a generalization of a known result of Sylvester, Kronecker and Zsigmondy on the prime factorization of Σn(e) (eϵZ). With Theorem 3.4., we improve corresponding results of Redei [11] and Sachs [14]. We generalize a technique developed in [8] and [3] and we study also the cases that K is a quadratic field and a cyclotomic field, respectively. Finally, we apply the results to the parameter determination of Fourier-like number-theoretic transforms in 𝔒K.
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