Field

A field is a commutative ring in which every nonzero element has a multiplicative inverse. Its operations support addition, subtraction, multiplication, and division by nonzero elements.

Ring (mathematics): A set with addition and multiplication satisfying specified algebraic laws, including distributivity. A field is a commutative ring with identity in which every nonzero element is invertible.
Rational number: A number expressible as a ratio of two integers with nonzero denominator. The rational numbers form the smallest field containing the integers.
Multiplicative inverse: For a nonzero element, an element whose product with it is one. The existence of these inverses is the defining property that distinguishes fields from general rings.
Vector space: A set with vector addition and scalar multiplication satisfying linearity axioms over a field. The field supplies the scalars used to combine vectors.
Integral domain: A commutative ring with identity and no zero divisors. Every field is an integral domain, but integral domains need not contain inverses for all nonzero elements.
Commutative ring: A ring whose multiplication is commutative. Commutativity is part of the ring structure required in the definition of a field.
Real number: An element of the complete ordered field extending the rational numbers. The real numbers provide the standard field for continuous mathematics.
Characteristic of a ring: The least positive number of repeated additions of one that yields zero, or zero if none exists. A field's characteristic is either zero or a prime number.
Linear algebra: The study of vector spaces, linear maps, and systems of linear equations. Its standard theory is formulated over fields.
Division ring: A ring in which every nonzero element has a multiplicative inverse, without requiring commutative multiplication. A field adds commutative multiplication to the division-ring axioms.

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