The author presents the relationships among: (1) the number of distinct values \(v\) of a polynomial \(f(x)\) of degree \(n\) over a finite field of \(q\) elements, (2) the degree \(u\) of the first non-vanishing elementary symmetric function of the values of \(f(x)\), and (3) the degree \(w\) of the first non-vanishing power sum of the values of \(f(x)\). The author provides an easy proof of the theorem of D. Wan: If \(v>q- (q- 1)/n\), then \(v=q\), i.e. \(f(x)\) is a permutation polynomial if \(v>q- (q- 1)/n\). The author also provides many other characterizations of permutation polynomials, as well as various examples and counterexamples.