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

Abstract

Arithmetic properties of some families in $\frac{{\mathbb{F}}_l[x]}{⟨x^{p^s{q}^t}-1⟩}$ are obtained by using the cyclotomic classes of order 2 with respect to $n=p^s{q}^t$, where $p\equiv 3\mathrm{mod}4$, $\gcd (\ddot{I}•(p^s),\ddot{I}•(q^t))=2$, $l$ is a primitive root modulo $q^t$ and ${\mathrm{ord}}_{p^s}(l)=\ddot{I}•(p^s)/2$. The form of these cyclotomic classes enables us to further generalize the results obtained in \cite{ref1}. The explicit expressions of primitive idempotents of minimal ideals in $\frac{{\mathbb{F}}_l[x]}{⟨x^{p^s{q}^t}-1⟩}$ are also obtained.