Safe Haskell	None
Language	Haskell2010

ExtensionField

Synopsis

data ExtensionField k im
class IrreducibleMonic k im where
- split :: ExtensionField k im -> Polynomial k
fromField :: ExtensionField k im -> [k]
fromList :: forall k im. (GaloisField k, IrreducibleMonic k im) => [k] -> ExtensionField k im
t :: Polynomial k -> Polynomial (ExtensionField k im)
x :: GaloisField k => Polynomial k

Documentation

data ExtensionField k im Source #

Extension fields GF(p^q)[X]/f(X) for p prime, q positive, and f(X) irreducible monic in GF(p^q)[X]

Instances

Eq k => Eq (ExtensionField k im) Source #
Instance details Defined in ExtensionField Methods (==) :: ExtensionField k im -> ExtensionField k im -> Bool # (/=) :: ExtensionField k im -> ExtensionField k im -> Bool #
(GaloisField k, IrreducibleMonic k im) => Fractional (ExtensionField k im) Source #	Extension fields are fields
Instance details Defined in ExtensionField Methods (/) :: ExtensionField k im -> ExtensionField k im -> ExtensionField k im # recip :: ExtensionField k im -> ExtensionField k im # fromRational :: Rational -> ExtensionField k im #
(GaloisField k, IrreducibleMonic k im) => Num (ExtensionField k im) Source #	Extension fields are rings
Instance details Defined in ExtensionField Methods (+) :: ExtensionField k im -> ExtensionField k im -> ExtensionField k im # (-) :: ExtensionField k im -> ExtensionField k im -> ExtensionField k im # (*) :: ExtensionField k im -> ExtensionField k im -> ExtensionField k im # negate :: ExtensionField k im -> ExtensionField k im # abs :: ExtensionField k im -> ExtensionField k im # signum :: ExtensionField k im -> ExtensionField k im # fromInteger :: Integer -> ExtensionField k im #
Show k => Show (ExtensionField k im) Source #
Instance details Defined in ExtensionField Methods showsPrec :: Int -> ExtensionField k im -> ShowS # show :: ExtensionField k im -> String # showList :: [ExtensionField k im] -> ShowS #
Generic (ExtensionField k im) Source #
Instance details Defined in ExtensionField Associated Types type Rep (ExtensionField k im) :: Type -> Type # Methods from :: ExtensionField k im -> Rep (ExtensionField k im) x # to :: Rep (ExtensionField k im) x -> ExtensionField k im #
(GaloisField k, IrreducibleMonic k im) => Random (ExtensionField k im) Source #	Extension fields are random
Instance details Defined in ExtensionField Methods randomR :: RandomGen g => (ExtensionField k im, ExtensionField k im) -> g -> (ExtensionField k im, g) # random :: RandomGen g => g -> (ExtensionField k im, g) # randomRs :: RandomGen g => (ExtensionField k im, ExtensionField k im) -> g -> [ExtensionField k im] # randoms :: RandomGen g => g -> [ExtensionField k im] # randomRIO :: (ExtensionField k im, ExtensionField k im) -> IO (ExtensionField k im) # randomIO :: IO (ExtensionField k im) #
(Arbitrary k, GaloisField k, IrreducibleMonic k im) => Arbitrary (ExtensionField k im) Source #	Extension fields are arbitrary
Instance details Defined in ExtensionField Methods arbitrary :: Gen (ExtensionField k im) # shrink :: ExtensionField k im -> [ExtensionField k im] #
NFData k => NFData (ExtensionField k im) Source #
Instance details Defined in ExtensionField Methods rnf :: ExtensionField k im -> () #
(GaloisField k, IrreducibleMonic k im) => Pretty (ExtensionField k im) Source #	Extension fields are pretty
Instance details Defined in ExtensionField Methods pretty :: ExtensionField k im -> Doc # prettyList :: [ExtensionField k im] -> Doc #
(GaloisField k, IrreducibleMonic k im) => GaloisField (ExtensionField k im) Source #	Extension fields are Galois fields
Instance details Defined in ExtensionField Methods char :: ExtensionField k im -> Integer Source # deg :: ExtensionField k im -> Int Source # order :: ExtensionField k im -> Integer Source # pow :: ExtensionField k im -> Integer -> ExtensionField k im Source # rnd :: MonadRandom m => m (ExtensionField k im) Source #
type Rep (ExtensionField k im) Source #
Instance details Defined in ExtensionField type Rep (ExtensionField k im)

class IrreducibleMonic k im where Source #

Irreducible monic splitting polynomial of extension field

Methods

split Source #

Arguments

:: ExtensionField k im
-> Polynomial k	Splitting polynomial

fromField :: ExtensionField k im -> [k] Source #

List from field

fromList :: forall k im. (GaloisField k, IrreducibleMonic k im) => [k] -> ExtensionField k im Source #

Field from list

t :: Polynomial k -> Polynomial (ExtensionField k im) Source #

Descend variable tower

x :: GaloisField k => Polynomial k Source #

Current indeterminate variable