We describe a simple procedure to find Aurifeuillian factors of values of cyclotomic polynomials ${\Phi }_d\left(a\right)$ for integers $a$ and $d\>0$. Assuming a suitable Riemann Hypothesis, the algorithm runs in deterministic time $\tilde{O}\left(d^{2}L\right)$, using $O\left(dL\right)$ space, where $L≔log(\left|a\right|+1)$.

Let $K$ be a number field defined by an irreducible polynomial $F\left(X\right)\in \mathbb{Z}\left[X\right]$ and ${\mathbb{Z}}_K$ its ring of integers. For every prime integer $p$, we give sufficient and necessary conditions on $F\left(X\right)$ that guarantee the existence of exactly $r$ prime ideals of ${\mathbb{Z}}_K$ lying above $p$, where $\overline{F}\left(X\right)$ factors into powers of $r$ monic irreducible polynomials in ${𝔽}_p\left[X\right]$. The given result presents a weaker condition than that given by S. K. Khanduja and M. Kumar (2010), which guarantees the existence of exactly $r$ prime ideals of ${\mathbb{Z}}_K$ lying above $p$. We further specify...