Complex Numbers form Field

Theorem
Consider the algebraic structure $\struct {\C, +, \times}$, where:


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

Then $\struct {\C, +, \times}$ forms a field.

Proof
From Complex Numbers under Addition form Abelian Group, we have that $\struct {\C, +}$ forms an abelian group.

From Non-Zero Complex Numbers under Multiplication form Abelian Group, we have that $\struct {\C_{\ne 0}, \times}$ forms an abelian group.

Finally, we have that Complex Multiplication Distributes over Addition.

Thus all the criteria are fulfilled, and $\struct {\C, +, \times}$ is a field.

Also see

 * Definition:Field of Complex Numbers