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BaseRing(A) : AlgQuat -> Fld
The base field of the quaternion algebra A.
The basis of the algebra A.
Discriminant(A) : AlgQuat[FldAlg] -> RngOrdIdl, SeqEnum
Given a quaternion algebra A returns the reduced discriminant,
the product of the primes which are ramified in A.
If A is defined over Q, it returns the positive
squarefree integer given
by the product of these primes. If A is defined over a number field,
it returns the ideal given by the product of ramified primes as well
as the sequence of real places where A is ramified.
RamifiedPlaces(A) : AlgQuat -> SeqEnum, SeqEnum
FactoredDiscriminant(A) : AlgQuat[FldRat] -> SeqEnum, SeqEnum
Given a quaternion algebra A over Q, returns the sequences of
finite ramified primes, i.e. those primes dividing the discriminant.
Note that the algebra is definite or indefinite, according to whether
the sequence is of odd or even length. The intrinsics RamifiedPlaces
and FactoredDiscriminant return as a second argument the sequence
which contains ∞ if A is definite.
The sequence of ramified primes of a quaternion algebra A over Q
determines the isomorphism class of the algebra.
> A := QuaternionAlgebra(-436,-503,22);
> RamifiedPrimes(A);
[ 17 ]
Provided the discriminant is of a size which can be factored, the
ramified primes are determined efficiently using Hilbert symbols.
FactoredDiscriminant(A) : AlgQuat[FldAlg] -> SeqEnum, SeqEnum
Given a quaternion algebra A over a number field F, returns the
sequence of finite ramified primes, i.e. those primes dividing the
discriminant, and the sequence of infinite ramified places.
Returns integers a and b in the base field F of the given quaternion algebra
A such that there exists i, j ∈A such that i2=a, j2=b, and ji= - ij.
The third object returned is the standard quaternion algebra
B= QuaternionAlgebra<F|a,b>,
and the fourth object is the homomorphism from A to B.
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