Copyright	(c) 2020 Federico Bongiorno
License	MIT
Maintainer	Federico Bongiorno <federicobongiorno97@gmail.com>
Safe Haskell	None
Language	Haskell2010

Math.NumberTheory.Moduli.Cbrt

Description

Cubic symbol of two Eisenstein Integers.

Synopsis

data CubicSymbol
- = Zero
- | Omega
- | OmegaSquare
- | One
cubicSymbol :: EisensteinInteger -> EisensteinInteger -> CubicSymbol
symbolToNum :: CubicSymbol -> EisensteinInteger

Documentation

data CubicSymbol Source #

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

Constructors

Zero
Omega
OmegaSquare
One

Instances

Instances details

Semigroup CubicSymbol Source #	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
Instance details Defined in Math.NumberTheory.Moduli.Cbrt Methods (<>) :: CubicSymbol -> CubicSymbol -> CubicSymbol sconcat :: NonEmpty CubicSymbol -> CubicSymbol stimes :: Integral b => b -> CubicSymbol -> CubicSymbol
Show CubicSymbol Source #
Instance details Defined in Math.NumberTheory.Moduli.Cbrt Methods showsPrec :: Int -> CubicSymbol -> ShowS show :: CubicSymbol -> String showList :: [CubicSymbol] -> ShowS
Eq CubicSymbol Source #
Instance details Defined in Math.NumberTheory.Moduli.Cbrt Methods (==) :: CubicSymbol -> CubicSymbol -> Bool (/=) :: CubicSymbol -> CubicSymbol -> Bool

cubicSymbol :: EisensteinInteger -> EisensteinInteger -> CubicSymbol Source #

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 Source #

Converts a cubic symbol to an Eisenstein Integer.