Algebra.Field.Finite

Documentation

data F p

Prime field of characteristic p. p should be prime, and not statically checked.

Instances

Reifies k p Integer => LeftModule Integer (F k p)

Methods

(.*) :: Integer -> F k p -> F k p

Reifies k p Integer => LeftModule Natural (F k p)

Methods

(.*) :: Natural -> F k p -> F k p

Reifies k p Integer => RightModule Integer (F k p)

Methods

(*.) :: F k p -> Integer -> F k p

Reifies k p Integer => RightModule Natural (F k p)

Methods

(*.) :: F k p -> Natural -> F k p

ConwayPolynomial p n => Reifies * (Conway Nat Nat p n) (Unipol (F Nat p))

Methods

reflect :: proxy (Unipol (F Nat p)) -> a

Eq (F k p)

Methods

(==) :: F k p -> F k p -> Bool

(/=) :: F k p -> F k p -> Bool

Reifies k p Integer => Fractional (F k p)

Methods

(/) :: F k p -> F k p -> F k p

recip :: F k p -> F k p

fromRational :: Rational -> F k p

Reifies k p Integer => Num (F k p)

Methods

(+) :: F k p -> F k p -> F k p

(-) :: F k p -> F k p -> F k p

(*) :: F k p -> F k p -> F k p

negate :: F k p -> F k p

abs :: F k p -> F k p

signum :: F k p -> F k p

fromInteger :: Integer -> F k p

Reifies k p Integer => Show (F k p)

Methods

showsPrec :: Int -> F k p -> ShowS

show :: F k p -> String

showList :: [F k p] -> ShowS

Reifies k p Integer => Random (F k p)

Methods

randomR :: RandomGen g => (F k p, F k p) -> g -> (F k p, g)

random :: RandomGen g => g -> (F k p, g)

randomRs :: RandomGen g => (F k p, F k p) -> g -> [F k p]

randoms :: RandomGen g => g -> [F k p]

randomRIO :: (F k p, F k p) -> IO (F k p)

randomIO :: IO (F k p)

Reifies k p Integer => IntegralDomain (F k p)

Methods

divides :: F k p -> F k p -> Bool

maybeQuot :: F k p -> F k p -> Maybe (F k p)

Reifies k p Integer => GCDDomain (F k p)

Methods

gcd :: F k p -> F k p -> F k p

reduceFraction :: F k p -> F k p -> (F k p, F k p)

lcm :: F k p -> F k p -> F k p

Reifies k p Integer => UFD (F k p)

Reifies k p Integer => PID (F k p)

Methods

egcd :: F k p -> F k p -> (F k p, F k p, F k p)

Reifies k p Integer => Euclidean (F k p)

Methods

degree :: F k p -> Maybe Natural

divide :: F k p -> F k p -> (F k p, F k p)

quot :: F k p -> F k p -> F k p

rem :: F k p -> F k p -> F k p

Reifies k p Integer => Commutative (F k p)

Reifies k p Integer => UnitNormalForm (F k p)

Methods

splitUnit :: F k p -> (F k p, F k p)

Reifies k p Integer => ZeroProductSemiring (F k p)

Reifies k p Integer => Ring (F k p)

Methods

fromInteger :: Integer -> F k p

Reifies k p Integer => Characteristic (F k p)

Methods

char :: proxy (F k p) -> Natural

Reifies k p Integer => Rig (F k p)

Methods

fromNatural :: Natural -> F k p

Reifies k p Integer => DecidableZero (F k p)

Methods

isZero :: F k p -> Bool

Reifies k p Integer => DecidableUnits (F k p)

Methods

recipUnit :: F k p -> Maybe (F k p)

isUnit :: F k p -> Bool

(^?) :: Integral n => F k p -> n -> Maybe (F k p)

Reifies k p Integer => DecidableAssociates (F k p)

Methods

isAssociate :: F k p -> F k p -> Bool

Reifies k p Integer => Division (F k p)

Methods

recip :: F k p -> F k p

(/) :: F k p -> F k p -> F k p

(\\) :: F k p -> F k p -> F k p

(^) :: Integral n => F k p -> n -> F k p

Reifies k p Integer => Unital (F k p)

Methods

one :: F k p

pow :: F k p -> Natural -> F k p

productWith :: Foldable f => (a -> F k p) -> f a -> F k p

Reifies k p Integer => Group (F k p)

Methods

(-) :: F k p -> F k p -> F k p

negate :: F k p -> F k p

subtract :: F k p -> F k p -> F k p

times :: Integral n => n -> F k p -> F k p

Reifies k p Integer => Multiplicative (F k p)

Methods

(*) :: F k p -> F k p -> F k p

pow1p :: F k p -> Natural -> F k p

productWith1 :: Foldable1 f => (a -> F k p) -> f a -> F k p

Reifies k p Integer => Semiring (F k p)

Reifies k p Integer => Monoidal (F k p)

Methods

zero :: F k p

sinnum :: Natural -> F k p -> F k p

sumWith :: Foldable f => (a -> F k p) -> f a -> F k p

Reifies k p Integer => Additive (F k p)

Methods

(+) :: F k p -> F k p -> F k p

sinnum1p :: Natural -> F k p -> F k p

sumWith1 :: Foldable1 f => (a -> F k p) -> f a -> F k p

Reifies k p Integer => Abelian (F k p)

NFData (F k p)

Methods

rnf :: F k p -> ()

Reifies k p Integer => Normed (F k p)

Associated Types

type Norm (F k p) :: *

Methods

norm :: F k p -> Norm (F k p)

liftNorm :: Norm (F k p) -> F k p

Reifies k p Integer => PrettyCoeff (F k p)

Methods

showsCoeff :: Int -> F k p -> ShowSCoeff

Reifies k p Integer => FiniteField (F k p)

Methods

power :: proxy (F k p) -> Natural

elements :: proxy (F k p) -> [F k p]

type Norm (F k p)

type Norm (F k p) = Integer

naturalRepr :: F p -> Integer

reifyPrimeField :: Integer -> (forall p. KnownNat p => Proxy (F p) -> a) -> a

withPrimeField :: Integer -> (forall p. KnownNat p => F p) -> Integer

modNat :: Reifies (p :: k) Integer => Integer -> F p

modNat' :: forall proxy p. Reifies p Integer => proxy (F p) -> Integer -> F p

modRat :: FiniteField k => Proxy k -> Fraction Integer -> k

modRat' :: FiniteField k => Fraction Integer -> k

class (Field k, Characteristic k) => FiniteField k where

Minimal complete definition

power, elements

Methods

power :: proxy k -> Natural

elements :: proxy k -> [k]

Instances

Reifies k p Integer => FiniteField (F k p)

Methods

power :: proxy (F k p) -> Natural

elements :: proxy (F k p) -> [F k p]

(KnownNat n, IsGF' * p n f) => FiniteField (GF' Nat p n f)

Methods

power :: proxy (GF' Nat p n f) -> Natural

elements :: proxy (GF' Nat p n f) -> [GF' Nat p n f]

order :: FiniteField k => proxy k -> Natural