Field (mathematics)

A field is a set with addition and multiplication in which every element has an additive inverse and every nonzero element has a multiplicative inverse, subject to familiar arithmetic laws.

Group: A set with a binary operation satisfying closure, associativity, an identity element, and inverses. Each field operation forms a group: addition on all elements and multiplication on nonzero elements.
Multiplicative inverse: An element that multiplies with a given element to produce the multiplicative identity. Every nonzero field element has a unique multiplicative inverse.
Galois field: A finite field, usually denoted GF(q), with q a prime power. Finite-field arithmetic underlies many error-correcting codes and cryptographic systems.
Integral domain: A commutative ring with identity and no zero divisors. Every field is an integral domain, but integral domains need not have inverses for all nonzero elements.
Commutative ring: A ring whose multiplication is commutative. A field is a commutative ring in which every nonzero element is invertible.
Characteristic of a ring: The least positive integer whose multiple of the multiplicative identity is zero, or zero if none exists. Characteristic records how repeated addition behaves inside a field.
Reed–Solomon code: An error-correcting code that represents messages by evaluations of polynomials over a finite field. Its encoding and recovery procedures use finite-field polynomial arithmetic.
Division ring: A ring in which every nonzero element has a multiplicative inverse, without requiring multiplication to be commutative. Fields require commutative multiplication; division rings do not.
Field axiom: A condition specifying the algebraic laws that addition and multiplication must satisfy in a field. These laws distinguish field arithmetic from operations on more general sets.
Prime subfield: The smallest subfield contained in a given field. It is generated by the identity and determines the field's characteristic.

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