For the field in relations, see field (relation).

A field is a set paired with two operations on the set, which are designated as addition ${\displaystyle (+)}$ and multiplication ${\displaystyle (\cdot)}$ . As a group can be conceptualized as an ordered pair of a set and an operation, ${\displaystyle (G,\cdot)}$ , a field can be conceptualized as an ordered triple ${\displaystyle (F,+,\cdot)}$ .

A set with addition and multiplication, ${\displaystyle (F,+,\cdot)}$ , is a field if and only if it satisfies the following properties:

1. Commutativity of both addition and multiplication — For all ${\displaystyle a, b \in F}$ , ${\displaystyle a + b = b + a}$ and ${\displaystyle a\cdot b=b\cdot a}$
2. Associativity of both addition and multiplication — For all ${\displaystyle a, b, c \in F}$ , ${\displaystyle (a+b)+c=a+(b+c)}$ and ${\displaystyle (a\cdot b)\cdot c=a\cdot(b\cdot c)}$
3. Additive Identity — There exists a "zero" element, ${\displaystyle 0\in F}$ , called an additive identity, such that ${\displaystyle a + 0 = a}$ for all ${\displaystyle a \in F}$
4. Additive Inverses — For each ${\displaystyle a \in F}$, there exists a ${\displaystyle b\in F}$ , called an additive inverse of ${\displaystyle a}$ , such that ${\displaystyle a+b=0}$
5. Multiplicative Identity — There exists a "one" element, ${\displaystyle 1\in F}$, different from 0, called a multiplicative identity, such that ${\displaystyle a \cdot 1=a}$ for all ${\displaystyle a \in F}$
6. Multiplicative Inverses — For each ${\displaystyle a \in F}$ , except for 0, there exists a ${\displaystyle c\in F}$ , called a multiplicative inverse of ${\displaystyle a}$ , such that ${\displaystyle a\cdot c=1}$
7. Distributive property — For all ${\displaystyle a, b, c \in F}$ , ${\displaystyle a\cdot(b+c)=a\cdot b+a\cdot c}$
8. Closure of addition and multiplication — For all ${\displaystyle a, b \in F}$ , ${\displaystyle a+b \in F}$ and ${\displaystyle a\cdot b \in F}$

Alternatively, a field can be defined as a commutative ring with unity (has a multiplicative identity) and multiplicative inverses.

We will often abbreviate the multiplication of two elements, ${\displaystyle a\cdot b}$ , by juxtaposition of the elements, ${\displaystyle ab}$ . Also, when combining multiplication and addition in an expression, multiplication takes precedence over addition unless the addition is enclosed in parenthesis. That is, ${\displaystyle a+bc+d=a+(bc)+d}$ .

We can also denote ${\displaystyle -a}$ and ${\displaystyle a^{-1} }$ as additive and multiplicative inverses of any ${\displaystyle a \in F}$ . Furthermore, we can define two more operations, called subtraction and division by ${\displaystyle a-b=a+(-b)}$ , and provided that ${\displaystyle b\neq 0}$ , ${\displaystyle \frac{a}{b}=a\cdot b^{-1}}$ .

Important Results

Because a field is also a ring with unity, these properties are inherited:

• ${\displaystyle (F,+)}$ is an abelian groups
• ${\displaystyle 0\cdot a=0}$ , for all ${\displaystyle a \in F}$
• ${\displaystyle a(-b)=(-a)b=-(ab)}$ , for all ${\displaystyle a, b \in F}$
• ${\displaystyle (-a)(-b)=ab}$ , for all ${\displaystyle a, b \in F}$
• ${\displaystyle (-1)\cdot a=-a}$ , for all ${\displaystyle a \in F}$
• ${\displaystyle (-1)\cdot(-1)=1}$
• Multiplication distributes over subtraction.

Additionally:

• ${\displaystyle (F^*,\cdot)}$ is also an abelian group, where ${\displaystyle F^*}$ is the set of nonzero elements of ${\displaystyle F}$
• Any field contains a subfield ${\displaystyle K}$ that is field-isomorphic to ${\displaystyle \mathbb{Q}}$ or ${\displaystyle \Z_{p}}$ for some prime ${\displaystyle p}$ .

Optional Properties

A field ${\displaystyle (F,+,\cdot)}$ is:

• A subfield of a field ${\displaystyle (K, +, \cdot)}$ if ${\displaystyle F\subseteq K}$ (see subset), where addition and multiplication on ${\displaystyle F}$ is a domain restriction on the addition and multiplication on ${\displaystyle K}$ . More commonly, we say that ${\displaystyle K}$ is an extension field of ${\displaystyle F}$ , and in fact, is also a vector space over ${\displaystyle F}$
• An ordered field if there exists a total order ${\displaystyle \le}$ on ${\displaystyle F}$ such that for all ${\displaystyle a,b,c \in G }$ , if ${\displaystyle a \le b}$, then ${\displaystyle a + c \le b + c}$ , (translation invariance), and if ${\displaystyle 0\le a}$ and ${\displaystyle 0 \le b}$ , then ${\displaystyle 0\le ab}$

Examples

• Under the usual operations of addition and multiplication, the rational numbers (${\displaystyle \mathbb{Q}}$), algebraic numbers (${\displaystyle \mathbb{A}}$), real numbers (${\displaystyle \R}$), and complex numbers (${\displaystyle \mathbb{C}}$) are fields.
• An extension field of ${\displaystyle \mathbb{Q}}$ , such as ${\displaystyle \Q\left[\sqrt2\right]=\left\{a+b\sqrt2\mid a,b\in\Q\right\}}$ .

Related

Elements of a field are the quantities over the vectorspaces are constructed and there are also called the scalars.

In the same branch functions ${\displaystyle X\to F}$ , where ${\displaystyle F}$ is a field are called scalar fields.