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 \({\mathbb{Q}}\) or \({\mathbb{F}}_q(X)\) with \(q\) odd, this function returns a list of primes or normalized irreducible polynomials corresponding to the finite ramified places of \(A\).

Example: Ramified Primes (ex-4e1f00)

The sequence of ramified primes of a quaternion algebra \(A\) over \({\mathbb{Q}}\) 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.

Run in calculator (opens in a new tab)Copy input

RamifiedPlaces(A): AlgQuat -> SeqEnum, SeqEnum

FactoredDiscriminant(A): AlgQuat -> SeqEnum, SeqEnum

Given a quaternion algebra \(A\) over \({\mathbb{Q}}\) or \({\mathbb{F}}_q(X)\) 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 \({\mathbb{Q}}\) or \({\mathbb{F}}_q(X)\).

Example: Ramified Primes over a Function Field (ex-642861)

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 ]

Run in calculator (opens in a new tab)Copy input

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 \(\Q\), \(\F_q(X)\) 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 \(i,j \in A\) where \(i^2=a\), \(j^2=b\), 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 \(B\).