  We study Poncelet's Theorem in finite projective coordinate planes over the
field $GF(p)$ and concentrate on a particular pencil of conics. For pairs of
such conics we investigate whether we can find polygons with $n$ sides, which
are inscribed in one conic and circumscribed about the other, so-called
Poncelet Polygons. By using suitable elements of the dihedral group for these
pairs, we prove that the length $n$ of such Poncelet Polygons is independent of
the starting point. In this sense Poncelet's Porism is valid. By using Euler's
divisor sum formula for the totient function, we can make a statement about the
number of different conic pairs, which carry Poncelet Polygons of length $n$.
Moreover, we will introduce polynomials whose zeros in $GF(p)$ yield
information about the relation of a given pair of conics. In particular, we can
decide for a given integer $n$, whether and how we can find Poncelet Polygons
for pairs of conics in the given coordinate plane. We will see that this
condition is closely connected with the theory of quadratic residues.
