# Fields in Algebra

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

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

**(F1):** is closed under addition, i.e. .

**(F2):** Associative Law holds in , i.e. for all .

**(F3):** Identity element with respect to addition exists in , i.e. there exist , such that , for all .

**(F4):** There exist inverses of every element of , i.e. for all , there exists an element such that .

**(F5):** Commutative Law holds in , i.e. for all .

**(F6):** is closed under multiplication, i.e. .

**(F7):** Associative Law holds in , i.e. for all .

**(F8):** Identity element with respect to multiplication exists in , i.e. there exists , such that , for all .

**(F9):** There exist inverses of every element of , i.e. for all and , there exists an element (multiplicative inverse) such that .

**(F10):** Commutative Law holds in , i.e. for all .

**(F11):** Distributive laws of multiplication over addition for all such that

and .

The above properties can be summarized as

**(1) ** is an abelian group.

**(2) ** is a semi-abelian group and is an abelian group.

**(3)** Multiplication is distributive over addition.