Cyclotomy, cyclotomic cosets and arithmetic properties of some families in $\frac{\mathbb{F}_l[x]}{\langle x^{p^sq^t}-1\rangle}$

TL;DR

This paper investigates the arithmetic properties of certain polynomial families over finite fields using cyclotomic classes, providing explicit primitive idempotents and generalizing previous results.

Contribution

It introduces new methods to analyze cyclotomic classes of order 2 and derives explicit primitive idempotents for minimal ideals in polynomial quotient rings.

Findings

Explicit expressions for primitive idempotents are obtained.

Generalization of previous cyclotomic class results.

Enhanced understanding of arithmetic properties in polynomial quotient rings.

Abstract

Arithmetic properties of some families in $\frac{\mathbb{F}_l[x]}{\langle x^{p^sq^t}-1\rangle}$ are obtained by using the cyclotomic classes of order 2 with respect to $n=p^sq^t$, where $p\equiv3 \mathrm{mod} 4$, $\gcd(\phi(p^s),\phi(q^t))=2$, $l$ is a primitive root modulo $q^t$ and $\mathrm{ord}_{p^s}(l)=\phi(p^s)/2$. 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 $\frac{\mathbb{F}_l[x]}{\langle x^{p^sq^t}-1\rangle}$ are also obtained.