HaskellForMaths-0.4.9: Combinatorics, group theory, commutative algebra, non-commutative algebra

Description

A module for arithmetic in quadratic number fields. A quadratic number field is a field of the form Q(sqrt d), where d is a square-free integer. For example, we can perform the following calculation in Q(sqrt 2):

(1 + sqrt 2) / (2 + sqrt 2)

It is also possible to mix different square roots in the same calculation. For example:

(1 + sqrt 2) * (1 + sqrt 3)

Square roots of negative numbers are also permitted. For example:

i * sqrt(-3)
Synopsis

# Documentation

data QNFBasis Source #

A basis for quadratic number fields Q(sqrt d), where d is a square-free integer.

Constructors

 One Sqrt Integer
Instances
 Source # Instance detailsDefined in Math.NumberTheory.QuadraticField Methods Source # Instance detailsDefined in Math.NumberTheory.QuadraticField Methods(/) :: QNF -> QNF -> QNF #recip :: QNF -> QNF # Source # Instance detailsDefined in Math.NumberTheory.QuadraticField Methods(<) :: QNFBasis -> QNFBasis -> Bool #(>) :: QNFBasis -> QNFBasis -> Bool # Source # Instance detailsDefined in Math.NumberTheory.QuadraticField MethodsshowList :: [QNFBasis] -> ShowS # (Eq k, Num k) => Algebra k QNFBasis Source # Instance detailsDefined in Math.NumberTheory.QuadraticField Methodsunit :: k -> Vect k QNFBasis Source #

The type for elements of quadratic number fields

Although this has the same name as the Prelude.sqrt function, it should be thought of as more like a constructor for creating elements of quadratic fields.

Note that for d positive, sqrt d means the positive square root, and sqrt (-d) should be interpreted as the square root with positive imaginary part, that is i * sqrt d. This has the consequence that for example, sqrt (-2) * sqrt (-3) = - sqrt 6.

newtype XVar Source #

Constructors

 X Int
Instances
 Source # Instance detailsDefined in Math.NumberTheory.QuadraticField Methods(==) :: XVar -> XVar -> Bool #(/=) :: XVar -> XVar -> Bool # Source # Instance detailsDefined in Math.NumberTheory.QuadraticField Methodscompare :: XVar -> XVar -> Ordering #(<) :: XVar -> XVar -> Bool #(<=) :: XVar -> XVar -> Bool #(>) :: XVar -> XVar -> Bool #(>=) :: XVar -> XVar -> Bool #max :: XVar -> XVar -> XVar #min :: XVar -> XVar -> XVar # Source # Instance detailsDefined in Math.NumberTheory.QuadraticField MethodsshowsPrec :: Int -> XVar -> ShowS #show :: XVar -> String #showList :: [XVar] -> ShowS #