Construction of a finite Dickson nearfield

For a Dickson pair $(q,n)$ we show that  $ \big \lbrace  \frac{q^k-1}{q-1}, 1 \leq k \leq  n \big  \rbrace $ forms a finite complete set of different residues modulo $n$. We also study the construction of a finite Dickson nearfield that arises from the Dickson pair $(q,n)$.