Explicit construction of an element of ${\rm GL}(2, p)$ of order $p+1$ It is well-known that the order of $GL(2, p)$ is $(p^2-1)(p^2-p) = (p-1)^2(p+1)p.$
It is easy to construct matrices of orders $(p-1)$ and $p$ (diagonal and parabolic, respectively), but the only way that leaps to mind of constructing an element of order $p+1$ is by constructing the companion matrix of the irreducible polynomial whose root is the multiplicative generator of $GF( p^2)$ and then raising it to $p-1$st power - this is quite non-explicit. I assume there is no explicit construction which works for all $p,$ but I could be wrong: is there?
 A: Here's a construction of sorts, which works for odd $p$ : find an element $d \in \mathbb{F}_{p}$ such that $-d$ is a non-square. Then the polynomial $x^{2}+d$ is irreducible in $\mathbb{F}_{p}[x]$. The matrices in $\{ \left( \begin{array}{clcr} a & b\\-db & a \end{array} \right) \}$ with $a, b \in \mathbb{F}_{p}$ (not both zero) form a cyclic group of order $p^{2}-1$. Admittedly, this is only very slightly more explicit than that outlined in the question.
A: Here is a slightly different way to think about this, though I admit that it is not really "explicit". Let $E$ be the field of order $p^2$, so $E$ is a two-dimensional vector space over the subfield $F$ of order $p$. The multiplicative group of $E$ is cyclic of order $p^2 - 1$, so if $x$ is a generator, then multiplication by $x$ is an $F$-linear transformation of $E$ with multiplicative order $p^2 - 1$. Now by choosing any $F$-basis for $E$, this linear transformation determines a matrix which is an element of $GL(2,p)$ having order $p^2 - 1$. A suitable power of this matrix has order $p+1$.
A: As the discussion shows, there really isn't any explicit construction which works for all primes $p$.   (Here you really want to specify that $p$ is odd, however, to avoid trivialities.)  It may or may not help to put the question into a somewhat broader framework; but anyway it's harmless to consider the same question for an arbitrary power $q$ of $p$.   
This begins historically with the work of Frobenius and Schur on characters of $G=\mathrm{SL}_2(\mathbb{F_q})$, followed by study of higher rank algebraic groups close to finite simple groups of Lie type (by many people including Brauer, Steinberg, Green, Macdonald, Deligne-Lusztig).   It's almost equivalent to study general or special linear groups here.  From the perspective of finite subgroups of semisimple algebraic groups over an algebraically closed field of characteristic $p$, one sees in the concrete case of $G$ that there are two distinct types of "maximal tori" over $\mathbb{F}_q$: the "split" diagonal group of order $q-1$ and the "anisotropic" (similar to "compact") group of order $q+1$.    Here the anisotropic torus becomes diagonal over a quadratic extension of $\mathbb{F}_q$.  (In general the types of finite tori are parametrized by something like the conjugacy classes in the Weyl group, here of order 2.)
Unfortunately there is no all-purpose recipe for a matrix of order $q+1$,
which over $\mathbb{F}_{q^2}$ is diagonalizable but not over $\mathbb{F}_q$: the best one can do using ordinary linear algebra is to specify the rational canonical form $$\begin{pmatrix} 0 & -1 \\ 1 & \theta + \theta^q \end{pmatrix},$$ where $\theta$ is a primitive $(q+1)$-th root of unity in the big field (so $\theta^{-1}= \theta^q$).   Here $\theta$ and its inverse are the eigenvalues of the matrix in question.
Some textbooks on characters of finite groups treat $G$ as an example, and there is a 2011 Springer text Representations of $\mathrm{SL}_2(\mathbb{F}_q)$ by Cedric Bonnafe which uses $G$ to introduce ideas from the Deligne-Lusztig theory.
