The complex numbers form a field

Builds on Complex numbers The reals form a field

Statement

The complex numbers $\mathbb{C}$ , with their addition and multiplication, form a field. That is, $\mathbb{C}$ is a commutative ring with $0 \neq 1$ , and each complex number $z \neq 0$ has a multiplicative inverse $z^{-1}$ with $z z^{-1} = 1$ .

Remarks

The inverse of a complex number $a + b i \neq 0$ is $\dfrac{a - b i}{a^2 + b^2}$ . The field $\mathbb{C}$ contains $\mathbb{R}$ as a subfield. It has dimension $2$ as a vector space over $\mathbb{R}$ , with basis $1, i$ . The field $\mathbb{C}$ is algebraically closed, but it is not an ordered field.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Complex numbers object
Real numbers object
The reals form a field statement
The complex numbers form a field statement

Where this leads

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Tag	`0FRSR10DWF`
URI	`m2://0FRSR10DWF`
Kind	statement, theorem
Writing	W0 stub
MSC	12D99
Status	This copy of the site has no build data. Run `m2 status 0FRSR10DWF`.
Uses	Complex numbers, The reals form a field

The complex numbers form a field

Statement

Remarks

Reading path

Where this leads

Further reading

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