ExtensionField

Documentation

data ExtensionField k im

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)

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)

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)

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)

Defined in ExtensionField

Methods

showsPrec :: Int -> ExtensionField k im -> ShowS

show :: ExtensionField k im -> String

showList :: [ExtensionField k im] -> ShowS

Generic (ExtensionField k im)

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)

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)

Defined in ExtensionField

Methods

arbitrary :: Gen (ExtensionField k im)

shrink :: ExtensionField k im -> [ExtensionField k im]

NFData k => NFData (ExtensionField k im)

Defined in ExtensionField

Methods

rnf :: ExtensionField k im -> ()

(GaloisField k, IrreducibleMonic k im) => Pretty (ExtensionField k im)

Defined in ExtensionField

Methods

pretty :: ExtensionField k im -> Doc

prettyList :: [ExtensionField k im] -> Doc

(GaloisField k, IrreducibleMonic k im) => GaloisField (ExtensionField k im)

Defined in ExtensionField

Methods

char :: ExtensionField k im -> Integer

deg :: ExtensionField k im -> Int

frob :: ExtensionField k im -> ExtensionField k im

order :: ExtensionField k im -> Integer

pow :: ExtensionField k im -> Integer -> ExtensionField k im

rnd :: MonadRandom m => m (ExtensionField k im)

type Rep (ExtensionField k im)

Defined in ExtensionField

type Rep (ExtensionField k im)

class IrreducibleMonic k im where

Irreducible monic splitting polynomial f(X) of extension field.

Methods

split :: ExtensionField k im -> Polynomial k

Splitting polynomial f(X).

fromField :: ExtensionField k im -> [k]

Convert from field element to list representation.

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

Convert from list representation to field element.

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

Descend tower of indeterminate variables.

x :: GaloisField k => Polynomial k

Current indeterminate variable.