Finite field arithmetic

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, finite field arithmetic is <a href="/facts/Arithmetic/hvHwkbMV">arithmetic</a> in a <a href="/facts/Finite_field/UkMGJo3m">finite field</a> (a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> containing a finite number of <a href="/facts/Element_(mathematics)/Q2mx1vHL">elements</a>) contrary to arithmetic in a field with an infinite number of elements, like the field of <a href="/facts/Rational_number/VJQmyY05">rational numbers</a>.
There are infinitely many different finite fields. Their <a href="/facts/Cardinality/m6VPojwT">number of elements</a> is necessarily of the form pn where p is a <a href="/facts/Prime_number/L20FLkgn">prime number</a> and n is a <a href="/facts/Positive_integer/u65og8JE">positive integer</a>, and two finite fields of the same size are <a href="/facts/Isomorphism/pj8KgkxU">isomorphic</a>. The prime p is called the <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a> of the field, and the positive integer n is called the <a href="/facts/Dimension_(vector_space)/z8m02A3W">dimension</a> of the field over its <a href="/facts/Characteristic_(algebra)/D2VzcQaG">prime field</a>.
Finite fields are used in a variety of applications, including in classical <a href="/facts/Coding_theory/SBqStoGL">coding theory</a> in <a href="/facts/Linear_block_code/QVuqpGga">linear block codes</a> such as <a href="/facts/BCH_code/1KelJGxF">BCH codes</a> and <a href="/facts/Reed%E2%80%93Solomon_error_correction/HPc3Qj6N">Reed–Solomon error correction</a>, in <a href="/facts/Cryptography/e94PEVau">cryptography</a> algorithms such as the <a href="/facts/Advanced_Encryption_Standard/HdXGBY2Q">Rijndael</a> (<a href="/facts/Advanced_Encryption_Standard/HdXGBY2Q">AES</a>) encryption algorithm, in tournament scheduling, and in the <a href="/facts/Design_of_experiments/VMrNftbr">design of experiments</a>.

Finite field arithmetic open-in-new

Finite field arithmetic