Draft of CubicSymbol

Math/NumberTheory/Moduli/CubicSymbol.hs

@@ -0,0 +1,102 @@
+{-# LANGUAGE LambdaCase #-}
+
+module Math.NumberTheory.Moduli.CubicSymbol
+    ( CubicSymbol(..)
+    , conj
+    , cubicSymbol
+    ) where
+
+import qualified Math.NumberTheory.Quadratic.EisensteinIntegers as E
+import qualified Math.NumberTheory.Primes as P
+import qualified Data.Euclidean as A
+import qualified Data.Semigroup as S
+import qualified Data.List as L
+
+
+
+data CubicSymbol = Zero | Omega | OmegaSquare | One deriving (Eq)
+
+instance S.Semigroup CubicSymbol where
+    (<>) = multiplication
+
+instance Show CubicSymbol where
+    show = \case
+        Zero         -> show 0
+        Omega        -> "ω"
+        OmegaSquare  -> "ω²"
+        One          -> show 1
+
+
+multiplication :: CubicSymbol -> CubicSymbol -> CubicSymbol
+multiplication Zero _ = Zero
+multiplication _ Zero = Zero
+multiplication One y = y
+multiplication x One = x
+multiplication Omega Omega = OmegaSquare
+multiplication Omega OmegaSquare = One
+multiplication OmegaSquare Omega = One
+multiplication OmegaSquare OmegaSquare = Omega
+
+
+
+exponentiation :: CubicSymbol -> Integer -> CubicSymbol
+exponentiation Zero _ = Zero
+exponentiation One _ = One
+exponentiation _ 0 = One
+exponentiation Omega 1 = Omega
+exponentiation Omega 2 = OmegaSquare
+exponentiation OmegaSquare 1 = OmegaSquare
+exponentiation OmegaSquare 2 = Omega
+
+
+
+conj :: CubicSymbol -> CubicSymbol
+conj = \case
+    Zero          -> Zero
+    Omega         -> OmegaSquare
+    OmegaSquare   -> Omega
+    One           -> One
+
+-- Converts Eisenstein integer to CubicSymbol value.
+toCubicSymbol :: E.EisensteinInteger -> CubicSymbol
+toCubicSymbol = \case
+    0               -> Zero
+    0 E.:+ 1        -> Omega
+    (-1) E.:+ (-1)  -> OmegaSquare
+    1               -> One
+
+
+
+
+cubicSymbol :: E.EisensteinInteger -> E.EisensteinInteger -> CubicSymbol
+cubicSymbol alpha n = Prelude.foldr (g) One factorisation
+    where factorisation = P.factorise n
+          g factor symbol = newSymbol <> symbol
+              where newSymbol = primePowerCubicSymbol alpha factor
+
+
+
+primePowerCubicSymbol :: E.EisensteinInteger -> (P.Prime E.EisensteinInteger, Word) -> CubicSymbol
+primePowerCubicSymbol alpha (primeNumber, wordExponent) = exponentiation symbol remainder
+    where remainder = exponent `rem` 3
+          exponent = Prelude.fromIntegral wordExponent
+          symbol =  primeCubicSymbol alpha prime
+          prime = P.unPrime primeNumber
+
+
+primeCubicSymbol :: E.EisensteinInteger -> E.EisensteinInteger -> CubicSymbol
+primeCubicSymbol alpha prime
+    -- Check whether @prime@ satifies norm and primality conditions.
+    | primeNorm == 3                 = error "This Eisenstein number is not coprime to 3."
+    -- Return 0 if @prime@ divides @alpha@
+    | otherwise                      = toCubicSymbol residue
+
+        -- A.rem returns the remainder of the Euclidean algorithm.
+        where primeNorm = E.norm prime
+              residue = Prelude.foldr (f) 1 listOfAlphas
+              f = \x y -> A.rem (x * y) prime
+              -- The function @take@ does not accept Integer values.
+              listOfAlphas = L.genericReplicate alphaExponent alpha
+              -- Exponent is defined to be 1/3*(@primeNorm@ - 1).
+              alphaExponent = primeNorm `div` 3
+

arithmoi.cabal

@@ -59,6 +59,7 @@ library
     Math.NumberTheory.Moduli
     Math.NumberTheory.Moduli.Chinese
     Math.NumberTheory.Moduli.Class
+    Math.NumberTheory.Moduli.CubicSymbol
     Math.NumberTheory.Moduli.DiscreteLogarithm
     Math.NumberTheory.Moduli.Equations
     Math.NumberTheory.Moduli.Jacobi