Editing Finite field arithmetic (section)

{{Short description|Arithmetic in a field with a finite number of elements}}
In [[mathematics]], '''finite field arithmetic''' is [[arithmetic]] in a [[finite field]] (a [[field (mathematics)|field]] containing a finite number of [[element (mathematics)|element]]s) contrary to arithmetic in a field with an infinite number of elements, like the field of [[rational number]]s.

There are infinitely many different finite fields. Their [[Cardinality|number of elements]] is necessarily of the form ''p<sup>n</sup>'' where ''p'' is a [[prime number]] and ''n'' is a [[positive integer]], and two finite fields of the same size are [[isomorphism|isomorphic]]. The prime ''p'' is called the [[characteristic (algebra)|characteristic]] of the field, and the positive integer ''n'' is called the [[dimension (vector space)|dimension]] of the field over its [[characteristic (algebra)#Case of fields|prime field]].

Finite fields are used in a variety of applications, including in classical [[coding theory]] in [[linear block code]]s such as [[BCH code]]s and [[Reed–Solomon error correction]], in [[cryptography]] algorithms such as the [[Advanced Encryption Standard|Rijndael]] ([[Advanced Encryption Standard|AES]]) encryption algorithm, in tournament scheduling, and in the [[design of experiments]].