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 {a^{ - 1}} and a' for a non-zero element a \in F. Let \left( 1 \right) be the unity of the field F.
\therefore \,a \cdot {a^{ - 1}} = 1 and a \cdot a' = 1 so that a \cdot {a^{ - 1}} = a \cdot a'. Since F - \left\{ 0 \right\} is a multiplicative group, applying left cancellation, we get {a^{  - 1}} = a'.

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 proving that s field is without zero divisors.
Let F be any field and let a,b \in F with a \ne 0 such that ab = 0. Let 1 be the unity of F. Since a  \ne 0, {a^{ - 1}} exists in Fand therefore

\begin{gathered} ab = 0 \Rightarrow {a^{ - 1}}\left( {ab}  \right) = {a^{ - 1}}0 \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow  \left( {{a^{ - 1}}a} \right)b = 0 \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow 1  \cdot b = 0 \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow b  = 0 \\ \end{gathered}


Similarly if b \ne 0 then it can be shown that ab = 0 \Rightarrow  a = 0.
Thus ab = 0  \Rightarrow a = 0\,\,\,{\text{or}}\,\,\,b = 0. Hence, a field is necessarily an integral domain.

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

Theorem 3: If a,b are any two elements of a field F and a \ne 0, there exists a unique element x such that a \cdot x = b.
Proof:
Let 1 be the unity of F and {a^{ - 1}}, the inverse of a in F then

a  \cdot \left( {{a^{ - 1}} \cdot b} \right) = \left( {a \cdot {a^{ - 1}}} \right)  \cdot b = 1 \cdot b = b


\begin{gathered} \therefore \,ax = b \Rightarrow a \cdot x = a  \cdot \left( {{a^{ - 1}}b} \right) \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,  \Rightarrow x = {a^{ - 1}}b \\ \end{gathered}


Thus, x = {a^{ - 1}}b  \in F.
Now, suppose there are two such elements {x_1},{x_2} (say) then a \cdot {x_1} = b and a \cdot {x_2} = b hence a \cdot {x_1} = a \cdot {x_2}. On applying left cancellation, we get {x_1} = {x_2}.
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.