Finite field

A finite field is a field with finitely many elements. Its size is always a power of a prime number, and fields of the same size are isomorphic.

Field (mathematics): A set with addition, subtraction, multiplication, and division by nonzero elements satisfying familiar arithmetic laws. Finite fields are precisely fields whose underlying sets have finitely many elements.
Quotient ring: A ring formed by identifying elements whose difference belongs to a specified ideal. A finite extension can be represented as polynomials modulo an irreducible polynomial.
Reed–Solomon code: An error-correcting code that represents messages as evaluations of polynomials over a finite field. Its symbols and polynomial evaluations are computed in a finite field.
Rational number: A number expressible as the ratio of two integers with a nonzero denominator. The rational numbers form an infinite field, unlike every finite field.
Prime number: A natural number greater than one with no positive divisors other than one and itself. Every finite field has characteristic equal to a prime.
Finite field extension: A finite field viewed as an extension of a smaller field, with finite vector-space dimension. Extensions build fields of prime-power size from their prime subfields.
Galois/Counter Mode: An authenticated-encryption mode combining counter encryption with polynomial-based authentication. Its authentication tag uses multiplication in the binary finite field GF(2^128).
Real number: An element of the complete ordered field extending the rational numbers. The real numbers illustrate an infinite field with an order, which finite fields cannot have.
Characteristic (algebra): The least positive number of copies of the multiplicative identity that sum to zero, or zero if none exists. A finite field’s characteristic determines the prime underlying its size.
Multiplicative group of a finite field: The group of nonzero elements of a finite field under multiplication. This group is cyclic, enabling arithmetic through powers of a generator.

Linked from 51 pages