Field in Algebra

A commutative ring with unity is called a field if its every non-zero elements possesses a multiple inverse.

Thus a ring R in which the elements of R different from O form an abelian group under multiplication is a field. Hence, a set F, having at least two distinct elements together with two operations  + and  \times is said to form a field if the following axioms are satisfied:

(F1): F is closed under addition. i.e. \forall a,b \in F \Rightarrow a + b \in F.

(F2): Associative Law holds in F. i.e. for all a,b,c \in F \Rightarrow \left( {a + b} \right) + c = a + \left( {b + c} \right).

(F3): Identity element with respect to addition exists in F. i.e. there exist 0 \in F, such that a + 0 = 0 + a = a, for all a \in F.

(F4): There exist inverse of every element of F, i.e. for all a \in F, there exist an element  - a \in F such that a + \left( { - a} \right) = - a + a = 0.

(F5): Commutative Law holds in F. i.e. for all a,b \in F \Rightarrow a + b = b + a.

(F6): F is closed under multiplication. i.e. \forall a,b \in F \Rightarrow a \cdot b \in F.

(F7): Associative Law holds in F. i.e. for all a,b,c \in F \Rightarrow \left( {a \cdot b} \right) \cdot c = a \cdot \left( {b \cdot c} \right).

(F8): Identity element with respect to multiplication exists in F. i.e. there exist 1 \in F, such that a \cdot 1 = 1 \cdot a = a, for all a \in F.

(F9): There exist inverse of every element of F, i.e. for all a \in F and a \ne 0, there exist an element {a^{ - 1}} \in F (multiplicative inverse) such that a \cdot {a^{ - 1}} = {a^{ - 1}} \cdot a = 1.

(F10): Commutative Law holds in F. i.e. for all a,b \in F \Rightarrow a \cdot b = b \cdot a.

(F11): Distributive laws of multiplication over addition for all a,b,c \in Fsuch that
a \cdot \left( {b + c} \right) = a \cdot b + a \cdot c and \left( {b + c} \right) \cdot a = b \cdot + c \cdot a.

The above properties can be summarized as

(1) \left( {F, + } \right) is an abelian group.

(2) \left( {F, \times } \right) is a semi-abelian group and \left( {F - \left\{ 0 \right\}, \times } \right) is an abelian group.

(3) Multiplication is distributive over addition.