text: Finite fields A field is an algebraic structure in which multiplication, addition, subtraction, and division are defined and satisfy the properties that multiplication is associative and every nonzero element has a multiplicative inverse. This implies that exponentiation with integer exponents is well-defined, except for nonpositive powers of . Common examples are the field of complex numbers, the real numbers and the rational numbers, considered earlier in this article, which are all infinite.
  A finite field is a field with a finite number of elements. This number of elements is either a prime number or a prime power; that is, it has the form  where  is a prime number, and  is a positive integer. For every such , there are fields with  elements. The fields with  elements are all isomorphic, which allows, in general, working as if there were only one field with  elements, denoted
  One has
  for every
  A primitive element in  is an element  such that the set of the  first powers of  (that is, ) equals the set of the nonzero elements of  There are  primitive elements in  where  is Euler's totient function.
  In  the freshman's dream identity
  is true for the exponent . As  in  It follows that the map
  is linear over  and is a field automorphism, called the Frobenius automorphism. If  the field  has  automorphisms, which are the  first powers (under composition) of . In other words, the Galois group of  is cyclic of order , generated by the Frobenius automorphism.