A prime $p$ is called a Wieferich prime if $2^{p-1}\equiv 1 \pmod {p^2}$. A monic polynomial $f(x)\in {\mathbb Z}\, [x]$ of degree $N\ge 2$ is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta ,\theta ^2,\ldots ,\theta ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta )$, where $f(\theta )=0$. We show that ${\mathcal F}_p(x):=x^{2p}+2x^{p}+2$ is monogenic if and only if $p$ is not a Wieferich prime.

An abelian monogenic polynomial $f(x)\in {\mathbb Z}[x]$ is a monic polynomial of degree $N\ge 2$ that is irreducible over ${\mathbb Q}$, such that the Galois group of $f(x)$ over ${\mathbb Q}$ is abelian, and $\{1,\theta ,\theta ^2,\ldots ,\theta ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta )$, where $f(\theta )=0$. We determine all abelian monogenic trinomials of the form $x^{2n}+ax^{n}+b$, where $n,a,b\in {\mathbb Z}$ with $n\ge 1$ and $ab\ne 0$.

Let $f(x)=x^{2p}+ax^p+b^p$, where p is a prime and $a,b\in {\mathbb Z}$ with $ab\ne 0$. If $f(x)$ is irreducible over ${\mathbb Q}$, we say that $f(x)$ is monogenic if $\{1,\theta ,\theta ^2,\ldots ,\theta ^{2p-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta )$, where $f(\theta )=0$. We give a characterisation of the monogenic trinomials $f(x)$ according to their Galois groups. These results extend prior investigations of the authors.

This note provides an alternative proof of a theorem by Li et al. [‘On the primitivity of some trinomials over finite fields’, Adv. Math. (China) 44(3) (2015), 387–393] regarding the nonprimitivity of the trinomial $x^{n}+ax+b$ over $\mathbb {F}_{q^{m}}$ under the condition $a^{n}b^{1-n}\in \mathbb {F}_{q^{u}}^{\ast }$ for some positive integer $u<m$. We extend this result to the trinomial $x^{n}+a^{k}x^{k}+b^{k}$, showing its nonprimitivity over $\mathbb {F}_{q^{m}}$ when $ a^{n}b^{k-n}\in \mathbb {F}_{q^{u}}^{\ast }$ for some positive integer $u<m$. While the existing proof relies on the theory of linear recurrences over finite fields, our approach is short and self-contained, requiring no prior knowledge of this area.

We prove a new bound on collinear triples in subgroups of prime finite fields and use it to give some new bounds on exponential sums with trinomials.

Let ${{X}^{5}}\,+\,aX\,+\,b\,\in \,Z\left[ X \right]$ have Galois group ${{D}_{5}}$ . Let $\theta $ be a root of ${{X}^{5}}\,+\,aX\,+\,b$ . An explicit formula is given for the discriminant of $Q\left( \theta\right)$ .

Our concern is with a particular problem which arises in connection with a discrete-time Markov chain model for a graded manpower system. In this model, the members of an organisation are classified into distinct classes. As time passes, they move from one class to another, or to the outside world, in a random way governed by fixed transition probabilities. In this paper, the emphasis is placed on evaluating exact values of the probabilities of attaining and maintaining a structure.