Finite field

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a finite field or Galois field (so-named in honor of <a href="/facts/%C3%89variste_Galois/T0BPM1zP">Évariste Galois</a>) is a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> that contains a finite number of <a href="/facts/Element_(mathematics)/Q2mx1vHL">elements</a>. As with any field, a finite field is a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are the <a href="/facts/Integers_mod_n/0wtRa5dU">integers mod 
 
 
 
 p
 
 
 {\displaystyle p}
 
</a> when 
 
 
 
 p
 
 
 {\displaystyle p}
 
 is a <a href="/facts/Prime_number/L20FLkgn">prime number</a>.
The order of a finite field is its number of elements, which is either a prime number or a <a href="/facts/Prime_power/R0vo3uYp">prime power</a>. For every prime number 
 
 
 
 p
 
 
 {\displaystyle p}
 
 and every positive integer 
 
 
 
 k
 
 
 {\displaystyle k}
 
 there are fields of order 
 
 
 
 
 p
 
 k
 
 
 
 
 {\displaystyle p^{k}}
 
. All finite fields of a given order are <a href="/facts/Isomorphism/pj8KgkxU">isomorphic</a>.
Finite fields are fundamental in a number of areas of mathematics and <a href="/facts/Computer_science/lN3pEyPz">computer science</a>, including <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, <a href="/facts/Galois_theory/54wscJfj">Galois theory</a>, <a href="/facts/Finite_geometry/GL0nnEk8">finite geometry</a>, <a href="/facts/Cryptography/e94PEVau">cryptography</a> and <a href="/facts/Coding_theory/SBqStoGL">coding theory</a>.

Finite field open-in-new

Finite field