Algebra.Structures.BezoutDomain

Representation of Bezout domains. That is non-Noetherian analogues of principal ideal domains. This means that all finitely generated ideals are principal.

Documentation

class IntegralDomain a => BezoutDomain a where

Bezout domains

Compute a principal ideal from another ideal. Also give witness that the principal ideal is equal to the first ideal.

toPrincipal <a_1,...,a_n> = (<a>,u_i,v_i) where

sum (u_i * a_i) = a

a_i = v_i * a

Methods

toPrincipal :: Ideal a -> (Ideal a, [a], [a])

Instances

(EuclideanDomain a, Eq a) => BezoutDomain a

propToPrincipal :: (BezoutDomain a, Eq a) => Ideal a -> Bool

Test that the generated ideal is principal.

propIsSameIdeal :: (BezoutDomain a, Eq a) => Ideal a -> Bool

Test that the generated ideal generate the same elements as the given.

propBezoutDomain :: (BezoutDomain a, Eq a) => Ideal a -> a -> a -> a -> Property

dividesB :: (BezoutDomain a, Eq a) => a -> a -> Bool

gcdB :: BezoutDomain a => a -> a -> a

intersectionB :: (BezoutDomain a, Eq a) => Ideal a -> Ideal a -> Ideal a

Intersection without witness.

intersectionBWitness :: (BezoutDomain a, Eq a) => Ideal a -> Ideal a -> (Ideal a, [[a]], [[a]])

Intersection of ideals with witness.

If one of the ideals is the zero ideal then the intersection is the zero ideal.

solveB :: (BezoutDomain a, Eq a) => Vector a -> Matrix a

Coherence of Bezout domains.

crt :: (BezoutDomain a, Eq a) => [a] -> [a] -> (a, a)

Chinese remainder theorem

Given a_1,...,a_n and m_1,...,m_n such that gcd(m_i,m_j) = 1. Let m = m_1*...*m_n compute a such that:

1. a = a_i (mod m_i)
2. If b is such that

b = a_i (mod m_i)

then a = b (mod m)

The function return (a,m).