As Nick has already pointed out, only the nonzero elements of a finite field can have a multiplicative order.
Exercise: In any field, a finite multiplicative group is cyclic.
Exercise: Let p be a prime number, and let K be a field of characteristic p. Show that, in K,
a) (x + y)^p = x^p + y^p for every x, y in K [the "Freshman's dream"]
b) If k is the field of p elements, show that x^p = x for every x in k.
c) If K is a field of characteristic p, f is a positive integer, and q = p^f, then (x + y)^q = x^q + y^q
Exercise: Let k be the field with p elements, f a positive integer, and K the field of q = p^f elements. Determine the number of elements in K having degree f over k.
