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BaseRing(A) : AlgQuat -> Fld
The base field of the quaternion algebra A.
The basis of the algebra A.
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.
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 -> 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).
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[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.
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.
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