Math.NumberTheory.Moduli.Cbrt

Cubic symbol of two Eisenstein Integers.

Documentation

data CubicSymbol

Represents the cubic residue character It is either 0, ω, ω² or 1.

Constructors

Zero

Omega

OmegaSquare

One

Instances

Eq CubicSymbol

Defined in Math.NumberTheory.Moduli.Cbrt

Methods

(==) :: CubicSymbol -> CubicSymbol -> Bool

(/=) :: CubicSymbol -> CubicSymbol -> Bool

Show CubicSymbol

Defined in Math.NumberTheory.Moduli.Cbrt

Methods

showsPrec :: Int -> CubicSymbol -> ShowS

show :: CubicSymbol -> String

showList :: [CubicSymbol] -> ShowS

Semigroup CubicSymbol

The set of cubic symbols form a semigroup. Note stimes is allowed to take non-positive values. In other words, the set of non-zero cubic symbols is regarded as a group.

>>> import Data.Semigroup
>>> stimes (-1) Omega
ω²
>>> stimes 0 Zero
1

Defined in Math.NumberTheory.Moduli.Cbrt

Methods

(<>) :: CubicSymbol -> CubicSymbol -> CubicSymbol

sconcat :: NonEmpty CubicSymbol -> CubicSymbol

stimes :: Integral b => b -> CubicSymbol -> CubicSymbol

cubicSymbol :: EisensteinInteger -> EisensteinInteger -> CubicSymbol

Cubic symbol of two Eisenstein Integers. The first argument is the numerator and the second argument is the denominator. The latter must be coprime to 3. This condition is checked.

If the arguments have a common factor, the result is Zero, otherwise it is either Omega, OmegaSquare or One.

>>> cubicSymbol (45 + 23*ω) (11 - 30*ω)
0
>>> cubicSymbol (31 - ω) (1 +10*ω)
ω

symbolToNum :: CubicSymbol -> EisensteinInteger

Converts a cubic symbol to an Eisenstein Integer.