  The study of solutions to polynomial equations over finite fields has a long
history in mathematics and is an interesting area of contemporary research. In
recent years the subject has found important applications in the modelling of
problems from applied mathematical fields such as signal analysis, system
theory, coding theory and cryptology. In this connection it is of interest to
know criteria for the existence of squares and other powers in arbitrary finite
fields. Making good use of polynomial division in polynomial rings over finite
fields, we have examined a classical criterion of Euler for squares in odd
prime fields, giving it a formulation which is apt for generalization to
arbitrary finite fields and powers. Our proof uses algebra rather than
classical number theory, which makes it convenient when presenting basic
methods of applied algebra in the classroom.
