Complex Numbers form Field

Theorem
Consider the algebraic structure $$\left({\C, +, \times}\right)$$, where:


 * $$\C$$ is the set of all complex numbers;
 * $$+$$ is the operation of complex addition;
 * $$\times$$ is the operation of complex multiplication;

Then $$\left({\C, +, \times}\right)$$ forms a field.

Proof

 * From Additive Group of Complex Numbers, we have that $$\left({\C, +}\right)$$ forms an abelian group.


 * From Multiplicative Group of Complex Numbers, we have that $$\left({\C^*, \times}\right)$$ forms an abelian group.


 * Finally, we have that Complex Multiplication Distributes over Addition.

Thus all the criteria are fulfilled, and $$\left({\C, +, \times}\right)$$ is a field.