Field

A **field** is a set on which addition, subtraction, multiplication, and division are defined, and behave as these operations do on the rational and real numbers.

The best-known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. There are many other fields, such as fields of rational functions, algebraic function fields, etc.

Hence, a field is an algebraic structure.

Note that a field does not require an order on the set, only that two operations are defined on

F

{\displaystyle F}

 that satisfy the field axioms (see definition). The real numbers are an *ordered field*, which requires additional structure.

Fields serve as foundational notions in several mathematical domains. This includes different branches of mathematical analysis, which are based on fields with additional structure. Basic theorems in analysis hinge on the structural properties of the field of real numbers. Most importantly for algebraic purposes, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra.

## Definition

Informally, a field is a set along with two operations defined on that set: an addition operation written as

a
b

{\displaystyle a+b}

, and a multiplication operation written as

a
⋅
b

{\displaystyle a\cdot b}

. These behave similarly as they behave for rational numbers and real numbers, including the existence of an additive inverse

−
a

{\displaystyle -a}

 for all elements, and of a multiplicative inverse

b

−
1

{\displaystyle b^{-1}}

 for every nonzero element. This allows one also to consider the so-called *inverse operations* of subtraction and division by defining:

a
−
b
:=
a
(
−
b
)

{\displaystyle a - b := a + (-b)}

, and

a

/

b
:=
a
⋅

b

−
1

{\displaystyle a / b := a \cdot b^{-1}}

.

Formally, a field is a set

F

{\displaystyle F}

 with two binary operations called addition and multiplication defined on

F

{\displaystyle F}

. A binary operation on

F

{\displaystyle F}

 is a mapping

F
×
F
↦
F

{\displaystyle F \times F \mapsto F}

 of the Cartesian product of

F

{\displaystyle F}

 to

F

{\displaystyle F}

. For addition and multiplication these are

(
a
,
b
)
↦
a
b

{\displaystyle (a, b) \mapsto a+b}

 and

(
a
,
b
)
↦
a
⋅
b

{\displaystyle (a,b) \mapsto a \cdot b}

 respectively. These operations are required to satisfy the following properties, referred to as *field axioms*:

∀
,
a
,
b
,
c
∈
F

{\displaystyle \forall, a, b,c \in F}

* Associativity of addition and multiplication:

a
(
b
c
# )
(
a
b
)
c

{\displaystyle a+(b+c)=(a+b)+c}

, and

a
⋅
(
b
⋅
c
# )
(
a
⋅
b
)
⋅
c

{\displaystyle a\cdot (b\cdot c)=(a\cdot b)\cdot c}

.
* Commutativity of addition and multiplication:

a
# b
b
a

{\displaystyle a + b = b + a}

, and

a
⋅
# b
b
⋅
a

{\displaystyle a\cdot b=b\cdot a}

.
* Additive and multiplicative identity: there exist two different elements

0
,
1
∈
F

{\displaystyle 0, 1 \in F}

 such that

a
# 0
a

{\displaystyle a + 0 = a}

 and

a
⋅
# 1
a

{\displaystyle a \cdot 1=a}

.
* Additive inverses:

∀
a
∈
F

{\displaystyle \forall a \in F}

,

∃
(
−
a
)
∈
F

{\displaystyle \exists (-a) \in F}

, called the *additive inverse*, such that

a
(
−
a
# )
0

{\displaystyle a + (-a) = 0 }

.
* Multiplicative inverses:

∀
a
∈
F

{\displaystyle \forall a \in F}

 such that

a
≠
0

{\displaystyle a\ne0}

,

∃
(

a

−
1

)
∈
F

{\displaystyle \exists (a^{-1}) \in F}

 called the *multiplicative inverse*, such that

a
⋅

a

−
1

=
1

{\displaystyle a \cdot a^{-1} = 1}

.
* Distributivity of multiplication over addition:

a
⋅
(
b
c
# )
(
a
⋅
b
)
(
a
⋅
c
)

{\displaystyle a \cdot (b + c) = (a \cdot b) + (a \cdot c)}

.

This may be summarized by saying: a field has two operations, called addition and multiplication; it is an 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. Even more summarized: a field is a commutative ring where

0
≠
1

{\displaystyle 0 \neq 1}

 and all nonzero elements are invertible under multiplication.