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.