# Field (mathematics) A **Field** is a *set* on which *addition, subtraction, multiplication, and division* are defined and behave as the corresponding operations of rational and real numbers do. A field is thus a fundamental *algebraic structure*. A few well known examples are: * The field of [rational numbers](https://en.wikipedia.org/wiki/Rational_number) * The field of [real numbers](https://en.wikipedia.org/wiki/Real_number), $\mathbb{R}$ * The field of complex numbers ### Operations on a field The operations on a field $\mathbb{F}$ are *addition*: $+ : \mathbb{F} \times \mathbb{F} \rightarrow \mathbb{F}$ And *multiplication*: $\times : \mathbb{F} \times \mathbb{F} \rightarrow \mathbb{F}$ ### Classic definition Formally, a field is a [set](https://en.wikipedia.org/wiki/Set_(mathematics) "Set (mathematics)") _F_ together with two [binary operations](https://en.wikipedia.org/wiki/Binary_operation "Binary operation") on F called _addition_ and _multiplication_.[\[1\]](https://en.wikipedia.org/wiki/Field_(mathematics)#cite_note-1) A binary operation on F is a mapping _F_ × _F_ → _F_, that is, a correspondence that associates with each ordered pair of elements of _F_ a uniquely determined element of F.[\[2\]](https://en.wikipedia.org/wiki/Field_(mathematics)#cite_note-2)[\[3\]](https://en.wikipedia.org/wiki/Field_(mathematics)#cite_note-3) The result of the addition of _a_ and _b_ is called the sum of _a_ and _b_, and is denoted _a_ + _b_. Similarly, the result of the multiplication of _a_ and _b_ is called the product of _a_ and _b_, and is denoted _ab_ or _a_ ⋅ _b_. These operations are required to satisfy the following properties, referred to as _[field axioms](https://en.wikipedia.org/wiki/Axiom#Non-logical_axioms "Axiom")_ (in these axioms, a, b, and c are arbitrary [elements](https://en.wikipedia.org/wiki/Element_(mathematics) "Element (mathematics)") of the field F): - [**Associativity**](https://en.wikipedia.org/wiki/Associativity "Associativity") of addition and multiplication: _a_ \+ (_b_ + _c_) = (_a_ + _b_) + _c_, and _a_ · (_b_ · _c_) = (_a_ · _b_) · _c_. - [**Commutativity**](https://en.wikipedia.org/wiki/Commutativity "Commutativity") of addition and multiplication: _a_ + _b_ \= _b_ + _a_, and _a_ · _b_ \= _b_ · _a_. - [**Additive**](https://en.wikipedia.org/wiki/Additive_identity "Additive identity") and [**multiplicative identity**](https://en.wikipedia.org/wiki/Multiplicative_identity "Multiplicative identity"): there exist two different elements 0 and 1 in _F_ such that _a_ \+ 0 = _a_ and _a_ · 1 = _a_. - [**Additive inverses**](https://en.wikipedia.org/wiki/Additive_inverse): for every _a_ in _F_, there exists an element in _F_, denoted −_a_, called the _additive inverse_ of _a_, such that _a_ \+ (−_a_) = 0. - [**Multiplicative inverses**](https://en.wikipedia.org/wiki/Multiplicative_inverse "Multiplicative inverse"): for every _a_ ≠ 0 in _F_, there exists an element in _F_, denoted by _a_−1 or 1/_a_, called the _multiplicative inverse_ of _a_, such that _a_ · _a_−1 \= 1. - [**Distributivity**](https://en.wikipedia.org/wiki/Distributivity "Distributivity") of multiplication over addition: _a_ · (_b_ + _c_) = (_a_ · _b_) + (_a_ · _c_). This may be summarized by saying: a field has two operations, called addition and multiplication; it is an [abelian group](https://en.wikipedia.org/wiki/Abelian_group "Abelian group") under addition with 0 as the additive identity; the nonzero elements are an abelian group under multiplication with 1 as the multiplicative identity; and multiplication distributes over addition. --- References: * [Field, wikipedia](https://en.wikipedia.org/wiki/Field_(mathematics))