Cyclotomy, cyclotomic cosets and arithmetic properties of some families in $\frac{{\mathbb{F}}_l[x]}{⟨x^{p^s{q}^t}-1⟩}$

Arithmetic properties of some families in ${\mathbb{F}}_l[x]/⟨x^{p^s{q}^t}-1⟩$ were obtained by using the cyclotomic classes of order 2 with respect to $n=p^s{q}^t$, where $p\equiv 3\mathrm{m}\mathrm{o}\mathrm{d}4$, $gcd(\varphi (p^s),\varphi (q^t))=2$, $l$ is a primitive root modulo $q^t$, and ${\mathrm{o}\mathrm{r}\mathrm{d}}_{p^s}(l)=\varphi (p^s)/2$. The form of these cyclotomic classes enabled us to further generalize the results obtained in ^[1]. The explicit expressions of primitive idempotents of minimal ideals in ${\mathbb{F}}_l[x]/⟨x^{p^s{q}^t}-1⟩$ were also obtained.