Update Field (refs #132)

src/Data/Field.purs

@@ -15,9 +15,27 @@ import Data.Semiring (class Semiring, add, mul, one, zero, (*), (+))
 
 -- | The `Field` class is for types that are (commutative) fields.
 -- |
--- | `Field`s are exactly `EuclideanRing` + `CommutativeRing` so this class
--- | exists as a convenience, so a single constraint can be used when field-like
--- | behaviour is expected.
-class (EuclideanRing a, CommutativeRing a) <= Field a
+-- | Mathematically, a field is a ring which is commutative and in which every
+-- | nonzero element has a multiplicative inverse; these conditions correspond
+-- | to the `CommutativeRing` and `DivisionRing` classes in PureScript
+-- | respectively. However, the `Field` class has `EuclideanRing` and
+-- | `DivisionRing` as superclasses, which seems like a stronger requirement
+-- | (since `CommutativeRing` is a superclass of `EuclideanRing`). In fact, it
+-- | is not stronger, since any type which has law-abiding `CommutativeRing`
+-- | and `DivisionRing` instances permits exactly one law-abiding
+-- | `EuclideanRing` instance. We use a `EuclideanRing` superclass here in
+-- | order to ensure that a `Field` constraint on a function permits you to use
+-- | `div` on that type, since `div` is a member of `EuclideanRing`.
+-- |
+-- | This class has no laws or members of its own; it exists as a convenience,
+-- | so a single constraint can be used when field-like behaviour is expected.
+-- |
+-- | This module also defines a single `Field` instance for any type which has
+-- | both `EuclideanRing` and `DivisionRing` instances. Any other instance
+-- | would overlap with this instance, so no other `Field` instances should be
+-- | defined in libraries. Instead, simply define `EuclideanRing` and
+-- | `DivisionRing` instances, and this will permit your type to be used with a
+-- | `Field` constraint.
+class (EuclideanRing a, DivisionRing a) <= Field a
 
-instance fieldNumber :: (EuclideanRing a, CommutativeRing a) => Field a
+instance field :: (EuclideanRing a, DivisionRing a) => Field a