# Fields in Algebra

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

Thus a ring $R$ in which the elements of $R$ are 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 inverses of every element of $F$, i.e. for all $a \in F$, there exists 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 exists $1 \in F$, such that $a \cdot 1 = 1 \cdot a = a$, for all $a \in F$.

(F9): There exist inverses of every element of $F$, i.e. for all $a \in F$ and $a \ne 0$, there exists 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 $a,b,c \in F$ such 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.