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Field

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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 IdentityThere 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\cdot1=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\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:

Examples

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.

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