Editing Finite field (section)

== Properties ==

A finite field is a finite set that is a [[field (mathematics)|field]]; this means that multiplication, addition, subtraction and division (excluding division by zero) are defined and satisfy the rules of arithmetic known as the [[field axioms]].<ref name=":0">{{Cite book |last=Dummit |first=David Steven |title=Abstract algebra |last2=Foote |first2=Richard M. |date=2004 |publisher=Wiley |isbn=978-0-471-43334-7 |edition=3rd |location=Hoboken, NJ}}</ref>

The number of elements of a finite field is called its ''order'' or, sometimes, its ''size''. A finite field of order <math>q</math> exists if and only if <math>q</math> is a [[prime power]] <math>p^k</math> (where <math>p</math> is a prime number and <math>k</math> is a positive integer). In a field of order <math>p^k</math>, adding <math>p</math> copies of any element always results in zero; that is, the [[characteristic (algebra)|characteristic]] of the field is <math>p</math>.<ref name=":0" />

For <math>q=p^k</math>, all fields of order <math>q</math> are [[isomorphic]] (see ''{{slink||Existence and uniqueness}}'' below).<ref name="moore"/> Moreover, a field cannot contain two different finite [[field extension|subfield]]s with the same order. One may therefore identify all finite fields with the same order, and they are unambiguously denoted <math>\mathbb{F}_{q}</math>, <math>\mathbf{F}_q</math> or <math>\mathrm{GF}(q)</math>, where the letters GF stand for "Galois field".<ref>This latter notation was introduced by [[E. H. Moore]] in an address given in 1893 at the International Mathematical Congress held in Chicago {{harvnb|Mullen|Panario|2013|loc = p.&nbsp;10}}.</ref>

In a finite field of order <math>q</math>, the [[polynomial]] <math>X^q-X</math> has all <math>q</math> elements of the finite field as [[root of a polynomial|root]]s.{{Citation needed|date=April 2025}} The non-zero elements of a finite field form a [[multiplicative group]]. This group is [[cyclic group|cyclic]], so all non-zero elements can be expressed as powers of a single element called a [[primitive element (finite field)|primitive element]] of the field. (In general there will be several primitive elements for a given field.)<ref name=":0" />

The simplest examples of finite fields are the fields of prime order: for each [[prime number]] <math>p</math>, the [[prime field]] of order <math>p</math> may be constructed as the [[modular arithmetic|integers modulo <math>p</math>]], <math>\mathbb{Z}/p\mathbb{Z}</math>.<ref name=":0" />

The elements of the prime field of order <math>p</math> may be represented by integers in the range <math>0,\ldots,p - 1</math>. The sum, the difference and the product are the [[Euclidean division|remainder of the division]] by <math>p</math> of the result of the corresponding integer operation.<ref name=":0" /> The multiplicative inverse of an element may be computed by using the extended Euclidean algorithm (see ''{{slink|Extended Euclidean algorithm|Modular integers}}'').{{Citation needed|date=April 2025}}

Let <math>F</math> be a finite field. For any element <math>x</math> in <math>F</math> and any [[integer]] <math>n</math>, denote by <math>n \cdot x</math> the sum of <math>n</math> copies of <math>x</math>. The least positive <math>n</math> such that <math>n \cdot 1 = 0</math> is the characteristic <math>p</math> of the field.<ref name=":0" /> This allows defining a multiplication <math>(k,x) \mapsto k \cdot x</math> of an element <math>k</math> of <math>\mathrm{GF}(p)</math> by an element <math>x</math> of <math>F</math> by choosing an integer representative for <math>k</math>. This multiplication makes <math>F</math> into a <math>\mathrm{GF}(p)</math>-[[vector space]].<ref name=":0" /> It follows that the number of elements of <math>F</math> is <math>p^n</math> for some integer <math>n</math>.<ref name=":0" />

The [[identity (mathematics)|identity]]
<math display="block" id="powersum">(x+y)^p=x^p+y^p</math>
(sometimes called the [[freshman's dream]]) is true in a field of characteristic <math>p</math>.  This follows from the [[binomial theorem]], as each [[binomial coefficient]] of the expansion of <math>(x+y)^p</math>, except the first and the last, is a multiple of <math>p</math>.{{Citation needed|date=April 2025}}

By [[Fermat's little theorem]], if <math>p</math> is a prime number and <math>x</math> is in the field <math>\mathrm{GF}(p)</math> then <math>x^p=x</math>.  This implies the equality
<math display="block">X^p-X=\prod_{a\in \mathrm{GF}(p)} (X-a)</math>
for polynomials over <math>\mathrm{GF}(p)</math>.  More generally, every element in <math>\mathrm{GF}(p^n)</math> satisfies the polynomial equation <math>x^{p^n}-x=0</math>.{{Citation needed|date=April 2025}}

Any finite [[field extension]] of a finite field is [[Separable extension|separable]] and simple.  That is, if <math>E</math> is a finite field and <math>F</math> is a subfield of <math>E</math>, then <math>E</math> is obtained from <math>F</math> by adjoining a single element whose [[Minimal polynomial (field theory)|minimal polynomial]] is [[Separable polynomial|separable]]. To use a piece of jargon, finite fields are [[perfect field|perfect]].<ref name=":0" />

A more general [[algebraic structure]] that satisfies all the other axioms of a field, but whose multiplication is not required to be [[Commutative property|commutative]], is called a [[division ring]] (or sometimes ''skew field''). By [[Wedderburn's little theorem]], any finite division ring is commutative, and hence is a finite field.<ref name=":0" />