Editing Exponentiation (section)

===Finite fields===
{{Main|Finite field}}
A [[field (mathematics)|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 {{math|0}}. Common examples are the field of [[complex number]]s, the [[real number]]s and the [[rational number]]s, considered earlier in this article, which are all [[infinite set|infinite]].

A ''finite field'' is a field with a [[finite set|finite number]] of elements. This number of elements is either a [[prime number]] or a [[prime power]]; that is, it has the form <math>q=p^k,</math> where {{mvar|p}} is a prime number, and {{mvar|k}} is a positive integer. For every such {{mvar|q}}, there are fields with {{mvar|q}} elements. The fields with {{mvar|q}} elements are all [[isomorphic]], which allows, in general, working as if there were only one field with {{mvar|q}} elements, denoted <math>\mathbb F_q.</math>

One has
: <math>x^q=x</math>
for every <math>x\in \mathbb F_q.</math>

A [[primitive element (finite field)|primitive element]] in <math>\mathbb F_q</math> is an element {{mvar|g}} such that the set of the {{math|''q'' − 1}} first powers of {{mvar|g}} (that is, <math>\{g^1=g, g^2, \ldots, g^{p-1}=g^0=1\}</math>) equals the set of the nonzero elements of <math>\mathbb F_q.</math> There are <math>\varphi (p-1)</math> primitive elements in <math>\mathbb F_q,</math> where <math>\varphi</math> is [[Euler's totient function]].

In <math>\mathbb F_q,</math> the [[freshman's dream]] identity
: <math>(x+y)^p = x^p+y^p</math>
is true for the exponent {{mvar|p}}. As <math>x^p=x</math> in <math>\mathbb F_q,</math> It follows that the map
: <math>\begin{align}
F\colon{} & \mathbb F_q \to \mathbb F_q\\
& x\mapsto x^p
\end{align}</math>
is [[linear map|linear]] over <math>\mathbb F_q,</math> and is a [[field automorphism]], called the [[Frobenius automorphism]]. If <math>q=p^k,</math> the field <math>\mathbb F_q</math> has {{mvar|k}} automorphisms, which are the {{mvar|k}} first powers (under [[function composition|composition]]) of {{mvar|F}}. In other words, the [[Galois group]] of <math>\mathbb F_q</math> is [[cyclic group|cyclic]] of order {{mvar|k}}, generated by the Frobenius automorphism.

The [[Diffie–Hellman key exchange]] is an application of exponentiation in finite fields that is widely used for [[secure communication]]s. It uses the fact that exponentiation is computationally inexpensive, whereas the inverse operation, the [[discrete logarithm]], is computationally expensive. More precisely, if {{mvar|g}} is a primitive element in <math>\mathbb F_q,</math> then <math>g^e</math> can be efficiently computed with [[exponentiation by squaring]] for any {{mvar|e}}, even if {{mvar|q}} is large, while there is no known computationally practical algorithm that allows retrieving {{mvar|e}} from <math>g^e</math> if {{mvar|q}} is sufficiently large.