Math 2.0

Commutative algebra Example

The rationals as the field of fractions of the integers

Builds on Field of fractionsThe ring of integers

Example

The field of fractions of the ring of integers ℤ\mathbb{Z} is the field ℚ\mathbb{Q} of rational numbers. Each element of Frac⁡(ℤ)\operatorname{Frac}(\mathbb{Z}) is a class a/ba/b with a,b∈ℤa, b \in \mathbb{Z} and b≠0b \neq 0. Two pairs give the same class, a/b=c/da/b = c/d, if and only if ad=bca d = b c. The map ℤ→ℚ\mathbb{Z} \to \mathbb{Q}, n↦n/1n \mapsto n/1, is the usual inclusion of the integers in the rationals.

Remarks

Every rational number has a unique form a/ba/b with b>0b > 0 and gcd⁡(a,b)=1\gcd(a, b) = 1. Every subfield of ℂ\mathbb{C} contains ℤ\mathbb{Z}, so it contains ℚ\mathbb{Q}, by the universal property of the field of fractions.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionZero divisor, definition, no evidence levelZero divisordefinitionThe ring of integers, example, no evidence levelThe ring of integersexampleField, definition, no evidence levelFielddefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionField of fractions, definition, no evidence levelField of fractionsdefinitionThe rationals as the field of fractions of the integers, example, no evidence levelThe rationals as the …example
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  1. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. Unit of a ring definition
  5. Zero divisor definition
  6. The ring of integers example
  7. Field definition
  8. Integral domain definition
  9. Field of fractions definition
  10. The rationals as the field of fractions of the integers example

Further reading

  1. Web https://en.wikipedia.org/wiki/Field_of_fractions, Examples
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WritingW0 stub
MSC13B30
Instance ofField of fractions
UsesField of fractions, The ring of integers

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