On a Number‐Theoretic Result of Sylvester‐Kronecker‐Zsigmondy
Gabriele Steidl, Manfred Tasche · Mathematische Nachrichten · 1989
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.