Complex Numbers form Field

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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 Infinite Abelian Group, we have that $\struct {\C, +}$ forms an abelian group.

From Non-Zero Complex Numbers under Multiplication form Infinite 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.

$\blacksquare$


Also see


Sources

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