We are interested in determining the bound of the average of the degrees of the irreducible characters whose degrees are not divisible by some prime $p$ that guarantees a finite group $G$ of odd order is $p$-nilpotent. There was found to be a bound which depends on the prime $p$. By further restricting the average by fixing a subfield $k$ of the complex numbers and then computing the average of the degrees of the irreducible characters whose degrees are not divisible by $p$ and have values in $k$, then a bound was obtained which depends on both $p$ and $k$. Examples are provided which show best possible bounds.

Let $f(x) \in \mathbb{Z}[x]$ be monic with $\mathrm{deg}(f) = N \geq 2$. Suppose that $f(x)$ is monogenic, and that $f(x)$ is the characteristic polynomial of the $N$th order linear recurrence sequence $\Upsilon_f := (U_n)_{n \geq 0}$ with initial conditions $$U_0 = U_1 = \cdots = U_{N−2} = 0 \mathrm{\hspace {1 pc}and\hspace {1 pc}} U_{N−1} = 1.$$ Let $\pi (m)$ denote the length of the period of $\Upsilon_f$ modulo the integer $m \geq 2$, where $\mathrm{gcd}(m, f(0)) = 1$. Let $p$ be a prime such that $f(x)$ is irreducible over $\mathbb{F}_p$ and $f(x^p)$ is irreducible over $\mathbb{Q}$. We prove that $f(x^p)$ is monogenic if and only if $\pi (p^2) \neq \pi(p)$, which provides a new and simple test for the monogenicity of $f(x^p)$. We also present some infinite families of such polynomials. This article extends previous work of the author.