Field

A FielD, briefly, is an algebraic system of elements, usually called numbers, in which the

operations of addition, subtraction, multiplication, and division (except division by zero) may be performed without leaving the system, the formal associative, commutative, and distributive rules hold, and the equations a=b+x and a=b*y have solutions, without restriction, provided in the latter case a not=0.

Definition: Given a SeT F and two BinaryOperations defined on F, called addition and multiplication, denoted "+" and "*" respectively. Then (F,+,*) is called a FielD if:

• 1). For all a,b belonging to (F,+,*), both a+b and a*b belong to (F,+,*).
• Closure of F under + and *.

• 2). For all a,b,c in (F,+,*), a+(b+c)= (a+b)+c
• and a*(b*c)=(a*b)*c.
• Both + and * are Associative in (F,+,*).

• 3). For all a,b belonging to (F,+,*), a+b=b+a
• and a*b=b*a.
• Both+ and * are commutative in (F,+,*).

• 4). For all a,b,c, belonging to (F,+,*), a*(b+c)= (a*b)+(a*c)
• and (a+b)*c=(a*c)+(b*c).
• The operation * is distributive over the operation + in (F,+,*).

• 5). For all a belonging to (F,+,*), there exists an element 0, such a+0=a
• and 0+a=a.
• Existence of an additive identity in (F,+,*).

• 6). For all a belonging to (F,+,*), there exists an element 1, such that a*1=a
• and 1*a=a.
• Existence of a multiplicative identity in (F,+,*).

• 7). For all a belonging to (F,+,*), there exists an element -a in (F,+,*), such that
• a+(-a)=0 and (-a)+a=0.
• Existence of an additive inverse for all a in (F,+,*).

• 8). For all a not=0 belonging to (F,*,+), there exists an element a^-1 in (F,+,*),
• such that a*(a^-1)=1 and (a^-1)*a=1.
• Existence of a multiplicative inverse for all a in (F,+,*).

Note that a FielD is just a RinG where all elements have multiplicative inverses, except 0 which cannot. Examples: Three Common FielDs.

*The Rational Numbers Q={all a/b: a, b belong to Z, the IntegerNumbers, and b not=0}.

*The RealNumbers= R. .

*The ComplexNumbers C={x: x=a+bi, where a,b belong to R, and i=sqrt(-1)}.

The concept of a FielD is of use, for example, in defining VectorS and MatriceS, two structures in LinearAlgebra, whose components can be elements of an arbitrary FielD.