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MAGMA Computational Algebra System

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Attributes of Quaternion Algebras

BaseField(A) : AlgQuat -> Fld
BaseRing(A) : AlgQuat -> Fld
The base field of the quaternion algebra A.
Basis(A) : AlgQuat -> SeqEnum
The basis of the algebra A.
Discriminant(A) : AlgQuat[FldRat] -> RngElt
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.
RamifiedPrimes(A) : AlgQuat -> SeqEnum
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.

Example AlgQuat_Ramified_Primes (H88E9)

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.
RamifiedPlaces(A) : AlgQuat[FldAlg] -> SeqEnum, SeqEnum
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.
StandardForm(A) : AlgQuat -> RngElt, RngElt, AlgQuat, Map
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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