  We provide a recipe to construct towers of fields producing high order
elements in $\mathrm{GF}(q,2^n)$, for odd $q$, and in $\mathrm{GF}(2,2 \cdot
3^n)$, for $n \ge 1$. These towers are obtained recursively by $x_{n}^2 + x_{n}
= v(x_{n - 1})$, for odd $q$, or $x_{n}^3 + x_{n} = v(x_{n - 1})$, for $q=2$,
where $v(x)$ is a polynomial of small degree over the prime field
$\mathrm{GF}(q,1)$ and $x_n$ belongs to the finite field extension
$\mathrm{GF}(q,2^n)$, for $q$ odd, or to $\mathrm{GF}(2,2\cdot 3^n)$. Several
examples are carried out and analysed numerically. The lower bounds of the
orders of the groups generated by $x_n$, or by the discriminant $\delta_n$ of
the polynomial, are similar to the ones obtained in [BCG+09], but we get better
numerical results in some cases.
