Fix comments

Fixes a typo for `Math.NumberTheory.EisensteinInteger.associates`
and rewords the comment for
`Math.NumberTheory.EisensteinInteger.primes`, as well as a few
other comments.

Math/NumberTheory/EisensteinIntegers.hs

@@ -55,7 +55,7 @@ import Math.NumberTheory.Utils.FromIntegral             (integerToNatural)
 
 infix 6 :+
 
--- |An Eisenstein integer is a + bω, where a and b are both integers.
+-- | An Eisenstein integer is a + bω, where a and b are both integers.
 data EisensteinInteger = (:+) { real :: !Integer, imag :: !Integer }
     deriving (Eq, Ord, Generic)
 
@@ -98,7 +98,7 @@ instance Num EisensteinInteger where
 ids :: [EisensteinInteger]
 ids = take 6 (iterate ((1 + ω) *) 1)
 
--- | Produce a list of an @EisensteinInteger@s' associates.
+-- | Produce a list of an @EisensteinInteger@'s associates.
 associates :: EisensteinInteger -> [EisensteinInteger]
 associates e = map (e *) ids
 
@@ -109,18 +109,18 @@ associates e = map (e *) ids
 primary :: EisensteinInteger -> EisensteinInteger
 primary = head . filter (\p -> p `modE` 3 == 2) . associates
 
--- |Simultaneous 'quot' and 'rem'.
+-- | Simultaneous 'quot' and 'rem'.
 quotRemE
     :: EisensteinInteger
     -> EisensteinInteger
     -> (EisensteinInteger, EisensteinInteger)
 quotRemE = divHelper quot
 
--- |Eisenstein integer division, truncating toward zero.
+-- | Eisenstein integer division, truncating toward zero.
 quotE :: EisensteinInteger -> EisensteinInteger -> EisensteinInteger
 n `quotE` d = q where (q,_) = quotRemE n d
 
--- |Eisenstein integer remainder, satisfying
+-- | Eisenstein integer remainder, satisfying
 --
 -- > (x `quotE` y)*y + (x `remE` y) == x
 remE :: EisensteinInteger -> EisensteinInteger -> EisensteinInteger
@@ -218,9 +218,11 @@ findPrime p = case Moduli.sqrtModMaybe (9*k*k - 1) (FieldCharacteristic (PrimeNa
         k :: Integer
         k = p `div` 6
 
--- | An infinite list of the Eisenstein primes. Uses primes in Z to exhaustively
--- generate all Eisenstein primes (all in the first sextant), in order of
--- ascending magnitude.
+-- | An infinite list of Eisenstein primes. Uses primes in Z to exhaustively
+-- generate all Eisenstein primes in order of ascending magnitude.
+-- * Every prime is in the first sextant, so the list contains no associates.
+-- * Eisenstein primes from the whole complex plane can be generated by
+-- applying @associates@ to each prime in this list.
 primes :: [EisensteinInteger]
 primes = (2 :+ 1) : mergeBy (comparing norm) l r
   where (leftPrimes, rightPrimes) = partition (\p -> p `mod` 3 == 2) Sieve.primes
@@ -246,7 +248,8 @@ primes = (2 :+ 1) : mergeBy (comparing norm) l r
 -- in Theorem 8.4 in Chapter 8, a way is given to express any Eisenstein
 -- integer @μ@ as @(-1)^a * ω^b * (1 - ω)^c * product [π_i^a_i | i <- [1..N]]@
 -- where @a, b, c, a_i@ are nonnegative integers, @N > 1@ is an integer and
--- @π_i@ are primary primes.
+-- @π_i@ are primary primes (for a primary Eisenstein prime @p@,
+-- @p ≡ 2 (modE 3)@, see @primary@ above).
 --
 -- * Aplying @norm@ to both sides of Theorem 8.4:
 --    @norm μ = norm ((-1)^a * ω^b * (1 - ω)^c * product [ π_i^a_i | i <- [1..N]])@
@@ -277,8 +280,8 @@ factorise g = concat $
                                       \ == " ++ show z ++ "but (1 - ω) is not\
                                       \ a prime factor of z."
       where
-        -- | Remove all @1 :+ (-1)@ (which is associated to @2 :+ 1@) factors from the
-        -- argument.
+        -- Remove all @1 :+ (-1)@ (which is associated to @2 :+ 1@) factors
+        -- from the argument.
         (q, r) = divModE z (2 :+ 1)
     go z (p, e) | p `mod` 3 == 2 =
                     let e' = e `quot` 2 in (z `quotI` (p ^ e'), [(p :+ 0, e')])