  Let $q=p^r$ be a prime power, and let $f(x)=x^m-\gs_{m-1}x^{m-1}-
>...-\gs_1x-\gs_0$ be an irreducible polynomial over the finite field $\GF(q)$
of size $q$. A zero $\xi$ of $f$ is called {\em nonstandard (of degree $m$)
over $\GF(q)$} if the recurrence relation $u_m=\gs_{m-1}u_{m-1} + ... +
\gs_1u_1+\gs_0u_0$ with characteristic polynomial $f$ can generate the powers
of $\xi$ in a nontrivial way, that is, with $u_0=1$ and $f(u_1)\neq 0$. In
2003, Brison and Nogueira asked for a characterisation of all nonstandard cases
in the case $m=2$, and solved this problem for $q$ a prime, and later for
$q=p^r$ with $r\leq4$.
  In this paper, we first show that classifying nonstandard finite field
elements is equivalent to classifying those cyclic codes over $\GF(q)$
generated by a single zero that posses extra permutation automorphisms.
  Apart from two sporadic examples of degree 11 over $\GF(2)$ and of degree 5
over $\GF(3)$, related to the Golay codes, there exist two classes of examples
of nonstandard finite field elements. One of these classes (type I) involves
irreducible polynomials $f$ of the form $f(x)=x^m-f_0$, and is well-understood.
The other class (type II) can be obtained from a primitive element in some
subfield by a process that we call extension and lifting. We will use the known
classification of the subgroups of $\PGL(2,q)$ in combination with a recent
result by Brison and Nogueira to show that a nonstandard element of degree two
over $\GF(q)$ necessarily is of type I or type II, thus solving completely the
classification problem for the case $m=2$.
