### Field

**Definition :-**A Ring

**( R , + , . )**which has at least two elements is called a field if

a) It is a commutative ring

b) It is a ring with unity

c) All non zero elements are inversible w . r. t . multiplication .

i.e. a is not equivalent to zero then b in R such that ab = ba = 1 ( unity of the ring ) then b= 1/a i.e. multiplicative inverse of a .

The integral domain and field are both commutative rings with unity and their third property is different , i.e. for I. D. it is a ring without zero divisors and for a field it is a ring in which all non zero elements are inversible.

**Alternative Definition :-**Combining the above properties we can give an alternate definition of a field as below .

A Ring

**( R , + , . )**with at least two elements is called a Field if its non zero elements from an abelian group under multiplication .

The condition R2 for a ring proves closure and associativity for multiplication , The ring is with unity shows the existence of multiplicative identity . The commutative property proves the character of an abelian group .

All non zero elements having their inverses prove the existence of inverses . Hence the above alternative definition .

Intuitively, a field is a set

*F*that is a commutative group with respect to two compatible operations, addition and multiplication, with "compatible" being formalized by

*distributivity,*and the caveat that the additive identity (0) has no multiplicative inverse (one cannot divide by 0).

The most common way to formalize this is by defining a

*field*as a set together with two operations, usually called

*addition*and

*multiplication*, and denoted by + and ·, respectively, such that the following axioms hold;

*subtraction*and

*division*are defined implicitly in terms of the inverse operations of addition and multiplication:

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*Closure*of*F*under addition and multiplication- For all
*a*,*b*in*F*, both*a*+*b*and*a*·*b*are in*F*(or more formally, + and · are binary operations on*F*). *Associativity*of addition and multiplication- For all
*a*,*b*, and*c*in*F*, the following equalities hold:*a*+ (*b*+*c*) = (*a*+*b*) +*c*and*a*· (*b*·*c*) = (*a*·*b*) ·*c*. *Commutativity*of addition and multiplication- For all
*a*and*b*in*F*, the following equalities hold:*a*+*b*=*b*+*a*and*a*·*b*=*b*·*a*. - Existence of additive and multiplicative
*identity elements* - There exists an element of
*F*, called the*additive identity*element and denoted by 0, such that for all*a*in*F*,*a*+ 0 =*a*. Likewise, there is an element, called the*multiplicative identity*element and denoted by 1, such that for all*a*in*F*,*a*· 1 =*a*. To exclude the trivial ring, the additive identity and the multiplicative identity are required to be distinct. - Existence of
*additive inverses*and*multiplicative inverses* - For every
*a*in*F*, there exists an element −*a*in*F*, such that*a*+ (−*a*) = 0. Similarly, for any*a*in*F*other than 0, there exists an element*a*^{−1}in*F*, such that*a*·*a*^{−1}= 1. (The elements*a*+ (−*b*) and*a*·*b*^{−1}are also denoted*a*−*b*and*a*/*b*, respectively.) In other words,*subtraction*and*division*operations exist. *Distributivity*of multiplication over addition- For all
*a*,*b*and*c*in*F*, the following equality holds:*a*· (*b*+*c*) = (*a*·*b*) + (*a*·*c*).

*F*, +, ·, −,

^{−1}, 0, 1〉; of type 〈2, 2, 1, 1, 0, 0〉, consisting of two abelian groups:

*F*under +, −, and 0;*F*\ {0} under ·,^{−1}, and 1, with 0 ≠ 1,

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