The rationals form a field

Builds on Rational numbers Ring of integers

Statement

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

Remarks

The inverse of $a / b$ with $a \neq 0$ is $b / a$ . The field $\mathbb{Q}$ is the smallest subfield of the real numbers. Each field of characteristic $0$ contains a copy of $\mathbb{Q}$ . The integers are not a field, and $\mathbb{Q}$ is the field of fractions of $\mathbb{Z}$ .

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Tag	`0N2WPCC00T`
URI	`m2://0N2WPCC00T`
Kind	statement, theorem
Writing	W0 stub
MSC	12E99
Status	This copy of the site has no build data. Run `m2 status 0N2WPCC00T`.
Uses	Rational numbers, Ring of integers

The rationals form a field

Statement

Remarks

Reading path

Where this leads

Further reading

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