Let $f\left(x\right)=xⁿ+k_{n-1}{x}^{n-1}+k_{n-2}{x}^{n-2}+\cdots +k₁x+k₀\in \mathbb{Z}\left[x\right]$, where $3\le k_{n-1}\le k_{n-2}\le \cdots \le k₁\le k₀\le 2k_{n-1}-3$. We show that f(x) and f(x²) are irreducible over ℚ. Moreover, the upper bound of $2k_{n-1}-3$ on the coefficients of f(x) is the best possible in this situation.

We prove a generalisation of the entropy formula for certain algebraic ${\mathbb{Z}}^d$-actions given in [2] and [4]. This formula expresses the entropy as the logarithm of the Mahler measure of a Laurent polynomial in d variables with integral coefficients. We replace the rational integers by the integers in a number field and examine the entropy of the corresponding dynamical system.

We give lower bounds for the Mahler measure of totally positive algebraic integers. These bounds depend on the degree and the discriminant. Our results improve earlier ones due to A. Schinzel. The proof uses an explicit auxiliary function in two variables.

The one-parameter family of polynomials $J_n(x,y)={\sum }_{j=0}^n\left(\genfrac{}{}{0pt}{}{y+j}{j}\right)x^j$ is a subfamily of the two-parameter family of Jacobi polynomials. We prove that for each $n\ge 6$, the polynomial $J_n(x,y_0)$ is irreducible over $\mathbb{Q}$ for all but finitely many $y_0\in \mathbb{Q}$. If $n$ is odd, then with the exception of a finite set of $y_0$, the Galois group of $J_n(x,y_0)$ is $S_n$; if $n$ is even, then the exceptional set is thin.

Fix an integer $\ell \ge 3$. Rikuna introduced a polynomial $r(x,t)$ defined over a function field $K\left(t\right)$ whose Galois group is cyclic of order $\ell$, where $K$ satisfies some mild hypotheses. In this paper we define the family of generalized Rikuna polynomials${\left\{r_n(x,t)\right\}}_{n\ge 1}$ of degree ${\ell }^n$. The $r_n(x,t)$ are constructed iteratively from the $r(x,t)$. We compute the Galois groups of the $r_n(x,t)$ for odd $\ell$ over an arbitrary base field and give applications to arithmetic dynamical systems.