Editing Exponentiation (section)

==Integer powers in algebra==
The definition of exponentiation with positive integer exponents as repeated multiplication may apply to any [[associative operation]] denoted as a multiplication.<ref group="nb">More generally, [[power associativity]] is sufficient for the definition.</ref> The definition of {{math|''x''<sup>0</sup>}} requires further the existence of a [[multiplicative identity]].<ref>{{cite book |author-last=Bourbaki |author-first=Nicolas |title=Algèbre |date=1970 |publisher=Springer|at=I.2}}</ref>

An [[algebraic structure]] consisting of a set together with an associative operation denoted multiplicatively, and a multiplicative identity denoted by {{math|1}} is a [[monoid]]. In such a monoid, exponentiation of an element {{mvar|x}} is defined inductively by
* <math>x^0 = 1,</math>
* <math>x^{n+1} = x x^n</math> for every nonnegative integer {{mvar|n}}.

If {{mvar|n}} is a negative integer, <math>x^n</math> is defined only if {{mvar|x}} has a [[multiplicative inverse]].<ref>{{cite book |author-last=Bloom |author-first=David M. |title=Linear Algebra and Geometry |date=1979 |isbn=978-0-521-29324-2 |page=[https://archive.org/details/linearalgebrageo0000bloo/page/45 45] |publisher=Cambridge University Press |url=https://archive.org/details/linearalgebrageo0000bloo |url-access=registration}}</ref> In this case, the inverse of {{mvar|x}} is denoted {{math|''x''<sup>−1</sup>}}, and {{math|''x''<sup>''n''</sup>}} is defined as <math>\left(x^{-1}\right)^{-n}.</math>

Exponentiation with integer exponents obeys the following laws, for {{mvar|x}} and {{mvar|y}} in the algebraic structure, and {{mvar|m}} and {{mvar|n}} integers:
: <math>\begin{align}
x^0&=1\\
x^{m+n}&=x^m x^n\\
(x^m)^n&=x^{mn}\\
(xy)^n&=x^n y^n \quad \text{if } xy=yx, \text{and, in particular, if the multiplication is commutative.}
\end{align}</math>

These definitions are widely used in many areas of mathematics, notably for [[group (mathematics)|groups]], [[ring (mathematics)|rings]], [[field (mathematics)|fields]], [[square matrix|square matrices]] (which form a ring). They apply also to [[function (mathematics)|functions]] from a [[set (mathematics)|set]] to itself, which form a monoid under [[function composition]]. This includes, as specific instances, [[geometric transformation]]s, and [[endomorphism]]s of any [[mathematical structure]].

When there are several operations that may be repeated, it is common to indicate the repeated operation by placing its symbol in the superscript, before the exponent. For example, if {{mvar|f}} is a [[real function]] whose valued can be multiplied, <math>f^n</math> denotes the exponentiation with respect of multiplication, and <math>f^{\circ n}</math> may denote exponentiation with respect of [[function composition]]. That is,
: <math>(f^n)(x)=(f(x))^n=f(x) \,f(x) \cdots f(x),</math>
and
: <math>(f^{\circ n})(x)=f(f(\cdots f(f(x))\cdots)).</math>
Commonly, <math>(f^n)(x)</math> is denoted <math>f(x)^n,</math> while <math>(f^{\circ n})(x)</math> is denoted <math>f^n(x).</math>

===In a group===
A [[multiplicative group]] is a set with as [[associative operation]] denoted as multiplication, that has an [[identity element]], and such that every element has an inverse.

So, if {{mvar|G}} is a group, <math>x^n</math> is defined for every <math>x\in G</math> and every integer {{mvar|n}}.

The set of all powers of an element of a group form a [[subgroup]]. A group (or subgroup) that consists of all powers of a specific element {{mvar|x}} is the [[cyclic group]] generated by {{mvar|x}}. If all the powers of {{mvar|x}} are distinct, the group is [[isomorphic]] to the [[additive group]] <math>\Z</math> of the integers. Otherwise, the cyclic group is [[finite group|finite]] (it has a finite number of elements), and its number of elements is the [[order (group theory)|order]] of {{mvar|x}}. If the order of {{mvar|x}} is {{mvar|n}}, then <math>x^n=x^0=1,</math> and the cyclic group generated by {{mvar|x}} consists of the {{mvar|n}} first powers of {{mvar|x}} (starting indifferently from the exponent {{math|0}} or {{math|1}}).

Order of elements play a fundamental role in [[group theory]]. For example, the order of an element in a finite group is always a divisor of the number of elements of the group (the ''order'' of the group). The possible orders of group elements are important in the study of the structure of a group (see [[Sylow theorems]]), and in the [[classification of finite simple groups]].

Superscript notation is also used for [[conjugacy class|conjugation]]; that is, {{math|1=''g''<sup>''h''</sup> = ''h''<sup>−1</sup>''gh''}}, where {{math|''g''}} and {{math|''h''}} are elements of a group. This notation cannot be confused with exponentiation, since the superscript is not an integer. The motivation of this notation is that conjugation obeys some of the laws of exponentiation, namely <math>(g^h)^k=g^{hk}</math> and <math>(gh)^k=g^kh^k.</math>

===In a ring===
In a [[ring (mathematics)|ring]], it may occur that some nonzero elements satisfy <math>x^n=0</math> for some integer {{mvar|n}}. Such an element is said to be [[nilpotent]]. In a [[commutative ring]], the nilpotent elements form an [[ideal (ring theory)|ideal]], called the [[nilradical of a ring|nilradical]] of the ring.

If the nilradical is reduced to the [[zero ideal]] (that is, if <math>x\neq 0</math> implies <math>x^n\neq 0</math> for every positive integer {{mvar|n}}), the commutative ring is said to be [[reduced ring|reduced]]. Reduced rings are important in [[algebraic geometry]], since the [[coordinate ring]] of an [[affine algebraic set]] is always a reduced ring.

More generally, given an ideal {{mvar|I}} in a commutative ring {{mvar|R}}, the set of the elements of {{mvar|R}} that have a power in {{mvar|I}} is an ideal, called the [[radical of an ideal|radical]] of {{mvar|I}}. The nilradical is the radical of the [[zero ideal]]. A [[radical ideal]] is an ideal that equals its own radical. In a [[polynomial ring]] <math>k[x_1, \ldots, x_n]</math> over a [[field (mathematics)|field]] {{mvar|k}}, an ideal is radical if and only if it is the set of all polynomials that are zero on an affine algebraic set (this is a consequence of [[Hilbert's Nullstellensatz]]).

===Matrices and linear operators===
If {{math|''A''}} is a square matrix, then the product of {{math|''A''}} with itself {{math|''n''}} times is called the [[matrix power]]. Also <math>A^0</math> is defined to be the identity matrix,<ref>Chapter 1, Elementary Linear Algebra, 8E, Howard Anton.</ref> and if {{math|''A''}} is invertible, then <math>A^{-n} = \left(A^{-1}\right)^n</math>.

Matrix powers appear often in the context of [[discrete dynamical system]]s, where the matrix {{math|''A''}} expresses a transition from a state vector {{math|''x''}} of some system to the next state {{math|''Ax''}} of the system.<ref>{{citation |last=Strang |first=Gilbert |title=Linear algebra and its applications |publisher=Brooks-Cole |date=1988 |edition=3rd|at=Chapter 5}}</ref> This is the standard interpretation of a [[Markov chain]], for example. Then <math>A^2x</math> is the state of the system after two time steps, and so forth: <math>A^nx</math> is the state of the system after {{math|''n''}} time steps. The matrix power <math>A^n</math> is the transition matrix between the state now and the state at a time {{math|''n''}} steps in the future. So computing matrix powers is equivalent to solving the evolution of the dynamical system. In many cases, matrix powers can be expediently computed by using [[eigenvalues and eigenvectors]].

Apart from matrices, more general [[linear operator]]s can also be exponentiated. An example is the [[derivative]] operator of calculus, <math>d/dx</math>, which is a linear operator acting on functions <math>f(x)</math> to give a new function <math>(d/dx)f(x) = f'(x)</math>. The {{math|''n''}}th power of the differentiation operator is the {{math|''n''}}th derivative:
: <math>\left(\frac{d}{dx}\right)^nf(x) = \frac{d^n}{dx^n}f(x) = f^{(n)}(x).</math>

These examples are for discrete exponents of linear operators, but in many circumstances it is also desirable to define powers of such operators with continuous exponents. This is the starting point of the mathematical theory of [[c0-semigroup|semigroups]].<ref>E. Hille, R. S. Phillips: ''Functional Analysis and Semi-Groups''. American Mathematical Society, 1975.</ref> Just as computing matrix powers with discrete exponents solves discrete dynamical systems, so does computing matrix powers with continuous exponents solve systems with continuous dynamics. Examples include approaches to solving the [[heat equation]], [[Schrödinger equation]], [[wave equation]], and other partial differential equations including a time evolution. The special case of exponentiating the derivative operator to a non-integer power is called the [[fractional derivative]] which, together with the [[fractional integral]], is one of the basic operations of the [[fractional calculus]].

===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.