Algebra.Field
 Contents Class Properties
Synopsis
class C a => C a where
 (/) :: a -> a -> a recip :: a -> a fromRational' :: Rational -> a (^-) :: a -> Integer -> a
fromRational :: C a => Rational -> a
propDivision :: (Eq a, C a, C a) => a -> a -> Property
propReciprocal :: (Eq a, C a, C a) => a -> Property
Class
 class C a => C a where Source

Field again corresponds to a commutative ring. Division is partially defined and satisfies

```    not (isZero b)  ==>  (a * b) / b === a
not (isZero a)  ==>  a * recip a === one
```

when it is defined. To safely call division, the program must take type-specific action; e.g., the following is appropriate in many cases:

``` safeRecip :: (Integral a, Eq a, Field.C a) => a -> Maybe a
safeRecip x =
let (q,r) = one `divMod` x
in  toMaybe (isZero r) q
```

Typical examples include rationals, the real numbers, and rational functions (ratios of polynomial functions). An instance should be typically declared only if most elements are invertible.

Actually, we have also used this type class for non-fields containing lots of units, e.g. residue classes with respect to non-primes and power series. So the restriction not (isZero a) must be better isUnit a.

Minimal definition: recip or (/)

Methods
 (/) :: a -> a -> a Source
 recip :: a -> a Source
 fromRational' :: Rational -> a Source
 (^-) :: a -> Integer -> a Source Instances
 C Double C Float C T C T Integral a => C (Ratio a) (Ord a, C a) => C (T a) C a => C (T a) C a => C (T a) C a => C (T a) (C a, C a) => C (T a) (C a, C a) => C (T a) C a => C (T a) (C a, C a) => C (T a) C a => C (T a) C a => C (T a) (Eq a, C a) => C (T a) C a => C (T a) (IsScalar u, C a) => C (T u a) C v => C (T a v) (Ord i, C a) => C (T i a) C v => C (T a v)
 fromRational :: C a => Rational -> a Source
Needed to work around shortcomings in GHC.
Properties
 propDivision :: (Eq a, C a, C a) => a -> a -> Property Source
the restriction on the divisor should be isUnit a instead of not (isZero a)
 propReciprocal :: (Eq a, C a, C a) => a -> Property Source