Field of Complex Numbers

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.

$\blacksquare$

Sources

• Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.1$: Example $2$
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 4.15$: Example $16$
• B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 1.2$: Ring Example $4$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 19$