## `Cat::QuotientField` -- the category of quotient fields

#### Introduction

``Cat::QuotientField`(R)` represents the category of quotient fields over `R`.

#### Generating the category

`Cat::QuotientField(R)`

#### Parameters

`R` | - | A domain which must be from the category `Cat::IntegralDomain`.

#### Categories

`Cat::Field`

, `Cat::Algebra(R)`,

if `R` has `Cat::OrderedSet` then
`Cat::OrderedSet`

#### Details

- A `Cat::QuotientField` is the field of fractions over the integral domain `R`.

#### Entries

characteristic

  The characteristic of this domain, which is is the same as that of `R`.

#### Method denom: return denominator

- `denom(dom x)`
- - Must return the denominator of `x`, which is an element of `R`.

#### Method numer: return numerator

- `numer(dom x)`
- - Must return the numerator of `x`, which is an element of `R`.

#### Method equal: test for equality

- `equal(dom x, dom y)`
- - Implements an equality test by testing if the cross-product of numerators and denominators are equal in `R`.

#### Method iszero: test for zero

- `iszero(dom x)`
- - Implements a test for zero by testing if the numerator is zero in `R`.

#### Method _less: test if element is less

- `_less(dom x, dom y)`
- - Implements an ordering if the domain `R` has the axiom `Cat::OrderedSet`: If `R` is ordered then this method implements an ordering which is given by the ordering of the cross-product of numerators and denominators in `R`.

#### Method retract: returns retracted element

- `retract(dom x)`
- - If `x` may be ``retracted'' to an element of `R` (i.e. if the factor `x` may be regarded as an element of `R`) this element is returned, otherwise `FAIL` is returned.
  - The default implementation uses the method `"_divide"` to divide numerator and denominator.

#### Changes

- no changes

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