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.
RamifiedPrimes(A) : AlgQuat -> SeqEnum
Given a quaternion algebra A over Q or Fq(X) with q odd, this function returns a list of primes or normalized irreducible polynomials corresponding to the finite ramified places of A.

Example AlgQuat_Ramified_Primes (H93E10)

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 -> SeqEnum, SeqEnum
FactoredDiscriminant(A) : AlgQuat -> SeqEnum, SeqEnum
Given a quaternion algebra A over Q or Fq(X) with q odd or a number field, this function returns the finite as well as infinite places where A is ramified.

Note: The first return value of these functions is always a list of ideals, even if the algebra is given over Q or Fq(X).

Example AlgQuat_Ramified_Primes_FqX (H93E11)

This example shows the (minor) difference between RamifiedPrimes and RamifiedPlaces. 
> F<x> := RationalFunctionField( GF(5) );
> A := QuaternionAlgebra< F | 2, x >;
> R<x>:= Integers(F);
> RamifiedPrimes(A);
[ x ]
> RamifiedPlaces(A);
[
    Ideal of Univariate Polynomial Ring in x over GF(5) generated by x
]
[ Infinity ]
Discriminant(A) : AlgQuat[FldRat] -> RngIntElt
Discriminant(A) : AlgQuat[FldFunRat] -> RngUPolElt
Discriminant(A) : AlgQuat[FldAlg] -> RngOrdIdl, SeqEnum
The reduced discriminant of a quaternion algebra A over Q, Fq(X) with q odd or a number field. In the first two cases, the functions return the product of the ramified primes. Over number fields, they return the product of the ramified prime ideals as well as the sequence of ramified infinite 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 elements i, j ∈A where 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.
V2.28, 13 July 2023