# Theorems of Field

__Theorem 1__**:**

The multiplicative inverse of a non-zero element of a field is unique.

__Proof__**:
**Let there be two multiplicative inverse and for a non-zero element . Let be the unity of the field .

and so that . Since is a multiplicative group, applying left cancellation, we get .

__Theorem 2__**:
**A field is necessarily an integral domain.

__Proof__**:**

Since a field is a commutative ring with unity, therefore, in order to show that every field is an integral domain we only need to prove that s field is without zero divisors.

Let be any field and let with such that . Let be the unity of . Since , exists in , therefore

Similarly if then it can be shown that .

Thus . Hence, a field is necessarily an integral domain.

__Corollary__**:**

Since the integral domain has no zero divisor and the field is necessarily an integral domain, therefore the field has no zero-divisor.

__Theorem 3__**:**

If are any two elements of a field and , there exists a unique element such that .

__Proof__**:**

Let be the unity of and , the inverse of in then

Thus, .

Now, suppose there are two such elements (say), then and hence . On applying left cancellation, we get .

Hence the uniqueness is established.

__Theorem 4__**:**

Every finite integral domain is a field, or a finite commutative ring with no zero divisor is a field.