Field

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 $(+)$ and multiplication $(\cdot)$ . As a group can be conceptualized as an ordered pair of a set and an operation, $(G,\cdot)$ , a field can be conceptualized as an ordered triple $(F,+,\cdot)$ .

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

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

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

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

Important Results

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

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

Additionally:

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

Optional Properties

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

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

Examples

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

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 $X\to F$ , where $F$ is a field are called scalar fields.