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:

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).

A field is therefore an algebraic structure 〈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,

with · distributing over +.