- 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:
- Commutativity of both addition and multiplication — For all $ a,b\in F $ , $ a+b=b+a $ and $ a\cdot b=b\cdot a $
- 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) $
- Additive Identity — There exists a "zero" element, $ 0\in F $ , called an additive identity, such that $ a+0=a $ for all $ a\in F $
- Additive Inverses — For each $ a\in F $, there exists a $ b\in F $ , called an additive inverse of $ a $ , such that $ a+b=0 $
- Multiplicative Identity — There exists a "one" element, $ 1\in F $, different from 0, called a multiplicative identity, such that $ a\cdot1=a $ for all $ a\in F $
- 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 $
- Distributive property — For all $ a,b,c\in F $ , $ a\cdot(b+c)=a\cdot b+a\cdot c $
- 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\ne0 $ , $ \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 $ \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 ($ \Q $), algebraic numbers ($ \mathbb A $), real numbers ($ \R $), and complex numbers ($ \C $) are fields.
- An extension field of $ \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.