Safe Haskell	None
Language	Haskell2010

Algebra.Field.Prime

Description

Prime fields

Synopsis

data F (p :: k)
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
- power :: proxy k -> Natural
- elements :: proxy k -> [k]
order :: FiniteField k => proxy k -> Natural

Documentation

data F (p :: k) Source #

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

Instances

Instances details

Reifies p Integer => RightModule Integer (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods (*.) :: F p -> Integer -> F p #
Reifies p Integer => RightModule Natural (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods (*.) :: F p -> Natural -> F p #
Reifies p Integer => LeftModule Integer (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods (.*) :: Integer -> F p -> F p #
Reifies p Integer => LeftModule Natural (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods (.*) :: Natural -> F p -> F p #
Eq (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods (==) :: F p -> F p -> Bool # (/=) :: F p -> F p -> Bool #
Reifies p Integer => Fractional (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods (/) :: F p -> F p -> F p # recip :: F p -> F p # fromRational :: Rational -> F p #
Reifies p Integer => Num (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods (+) :: F p -> F p -> F p # (-) :: F p -> F p -> F p # (*) :: F p -> F p -> F p # negate :: F p -> F p # abs :: F p -> F p # signum :: F p -> F p # fromInteger :: Integer -> F p #
Ord (F p) Source #	Caution: just for use with Map or Sets; no guarantee for the compatibility with field structure and normal forms!
Instance details Defined in Algebra.Field.Prime Methods compare :: F p -> F p -> Ordering # (<) :: F p -> F p -> Bool # (<=) :: F p -> F p -> Bool # (>) :: F p -> F p -> Bool # (>=) :: F p -> F p -> Bool # max :: F p -> F p -> F p # min :: F p -> F p -> F p #
Reifies p Integer => Read (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods readsPrec :: Int -> ReadS (F p) # readList :: ReadS [F p] # readPrec :: ReadPrec (F p) # readListPrec :: ReadPrec [F p] #
Reifies p Integer => Show (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods showsPrec :: Int -> F p -> ShowS # show :: F p -> String # showList :: [F p] -> ShowS #
Reifies p Integer => UFD (F p) Source #
Instance details Defined in Algebra.Field.Prime
Reifies p Integer => PID (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods egcd :: F p -> F p -> (F p, F p, F p) #
Reifies p Integer => GCDDomain (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods gcd :: F p -> F p -> F p # reduceFraction :: F p -> F p -> (F p, F p) # lcm :: F p -> F p -> F p #
Reifies p Integer => IntegralDomain (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods divides :: F p -> F p -> Bool # maybeQuot :: F p -> F p -> Maybe (F p) #
Reifies p Integer => Euclidean (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods degree :: F p -> Maybe Natural # divide :: F p -> F p -> (F p, F p) # quot :: F p -> F p -> F p # rem :: F p -> F p -> F p #
Reifies p Integer => ZeroProductSemiring (F p) Source #
Instance details Defined in Algebra.Field.Prime
Reifies p Integer => UnitNormalForm (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods splitUnit :: F p -> (F p, F p) #
Reifies p Integer => Ring (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods fromInteger :: Integer -> F p
Reifies p Integer => Rig (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods fromNatural :: Natural -> F p #
Reifies p Integer => Characteristic (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods char :: proxy (F p) -> Natural #
Reifies p Integer => DecidableZero (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods isZero :: F p -> Bool #
Reifies p Integer => DecidableUnits (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods recipUnit :: F p -> Maybe (F p) # isUnit :: F p -> Bool # (^?) :: Integral n => F p -> n -> Maybe (F p) #
Reifies p Integer => DecidableAssociates (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods isAssociate :: F p -> F p -> Bool #
Reifies p Integer => Unital (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods one :: F p # pow :: F p -> Natural -> F p # productWith :: Foldable f => (a -> F p) -> f a -> F p #
Reifies p Integer => Division (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods recip :: F p -> F p # (/) :: F p -> F p -> F p # (\\) :: F p -> F p -> F p # (^) :: Integral n => F p -> n -> F p
Reifies p Integer => Commutative (F p) Source #
Instance details Defined in Algebra.Field.Prime
Reifies p Integer => Semiring (F p) Source #
Instance details Defined in Algebra.Field.Prime
Reifies p Integer => Multiplicative (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods (*) :: F p -> F p -> F p # pow1p :: F p -> Natural -> F p # productWith1 :: Foldable1 f => (a -> F p) -> f a -> F p #
Reifies p Integer => Monoidal (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods zero :: F p # sinnum :: Natural -> F p -> F p # sumWith :: Foldable f => (a -> F p) -> f a -> F p #
Reifies p Integer => Group (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods (-) :: F p -> F p -> F p # negate :: F p -> F p # subtract :: F p -> F p -> F p # times :: Integral n => n -> F p -> F p #
Reifies p Integer => Additive (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods (+) :: F p -> F p -> F p # sinnum1p :: Natural -> F p -> F p # sumWith1 :: Foldable1 f => (a -> F p) -> f a -> F p #
Reifies p Integer => Abelian (F p) Source #
Instance details Defined in Algebra.Field.Prime
Hashable (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods hashWithSalt :: Int -> F p -> Int # hash :: F p -> Int #
NFData (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods rnf :: F p -> () #
Reifies p Integer => Random (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods randomR :: RandomGen g => (F p, F p) -> g -> (F p, g) random :: RandomGen g => g -> (F p, g) randomRs :: RandomGen g => (F p, F p) -> g -> [F p] randoms :: RandomGen g => g -> [F p]
Reifies p Integer => FiniteField (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods power :: proxy (F p) -> Natural Source # elements :: proxy (F p) -> [F p] Source #
Reifies p Integer => Normed (F p) Source #
Instance details Defined in Algebra.Field.Prime Associated Types type Norm (F p) Source # Methods norm :: F p -> Norm (F p) Source # liftNorm :: Norm (F p) -> F p Source #
Reifies p Integer => PrettyCoeff (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods showsCoeff :: Int -> F p -> ShowSCoeff Source #
type Norm (F p) Source #
Instance details Defined in Algebra.Field.Prime type Norm (F p) = Integer

naturalRepr :: F p -> Integer Source #

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

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

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

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

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

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

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

Abstract class for finite fields.

Methods

power :: proxy k -> Natural Source #

elements :: proxy k -> [k] Source #

Instances

Instances details

Reifies p Integer => FiniteField (F p) Source #
Instance details Defined in Algebra.Field.Prime Methods power :: proxy (F p) -> Natural Source # elements :: proxy (F p) -> [F p] Source #

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