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 Complex Numbers under Addition form Abelian Group, we have that $\left({\C, +}\right)$ forms an abelian group.


 * From Non-Zero Complex Numbers under Multiplication form Abelian Group, we have that $\left({\C_{\ne 0}, \times}\right)$ 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.