Elements of Quaternion Algebras

For more information about elements of orders of associative algebras, see Section Orders.

Creation of Elements

A ! 0: AlgQuat, RngIntElt -> AlgQuatElt

Zero(A): AlgQuat -> AlgQuatElt

The zero element of the quaternion algebra \(A\).

A ! 1: AlgQuat, RngIntElt -> AlgQuatElt

One(A): AlgQuat -> AlgQuatElt

The identity element of the quaternion algebra \(A\).

A . i: AlgQuat, RngIntElt -> AlgQuatElt

Name(A, i): AlgQuat, RngIntElt -> AlgQuatElt

Given a quaternion algebra \(A\) and an integer \(1\le i\le 3\), returns the \(i\)th generator of \(A\) as an algebra over the base ring. Note that the element \(1\) is always the first element of a basis, and is never returned as a generating element.

A ! x: AlgQuat, Any -> AlgQuatElt

Return an element of the quaternion algebra \(A\) described by \(x\), where \(x\) may be an algebra element, a module element, a sequence, an element of an order of an associative algebra or be coercible into the coefficient ring of \(A\).

Arithmetic of Elements

x + y: AlgQuatElt, AlgQuatElt -> AlgQuatElt

The sum of \(x\) and \(y\).

x - y: AlgQuatElt, AlgQuatElt -> AlgQuatElt

The difference of \(x\) and \(y\).

x * y: AlgQuatElt, AlgQuatElt -> AlgQuatElt

The product of \(x\) and \(y\).

x / y: AlgQuatElt, AlgQuatElt -> AlgQuatElt

x / y: AlgQuatOrdElt, AlgQuatOrdElt -> AlgQuatElt

The quotient of \(x\) by the unit \(y\) in the quaternion algebra.

x eq y: AlgQuatElt, AlgQuatElt -> BoolElt

Returns true if the elements \(x\) and \(y\) are equal; otherwise false.

x ne y: AlgQuatElt, AlgQuatElt -> BoolElt

Returns true if and only if the elements \(x\) and \(y\) are not equal.

x in A: AlgQuatElt, AlgQuat -> BoolElt

Returns true if and only if \(x\) is in the algebra \(A\).

x notin A: AlgQuatElt, AlgQuat -> BoolElt

Returns true if and only if \(x\) is not in the algebra \(A\).

Conjugate(x): AlgQuatElt -> AlgQuatElt

Conjugate(x): AlgQuatOrdElt -> AlgQuatOrdElt

Conjugate(x): AlgAssVOrdElt -> AlgAssVOrdElt

The conjugate \(\bar{x}\) of the element \(x\) of a quaternion algebra, defined so that the reduced trace and reduced norm are \(\bar{x}+x\) and \(\bar{x}x\), respectively.

ElementToSequence(x): AlgQuatElt -> SeqEnum

Eltseq(x): AlgQuatElt -> SeqEnum

Coordinates(x): AlgQuatElt -> SeqEnum

Given an element \(x\) of a quaternion algebra or order, this function returns the sequence of coordinates of \(x\) in terms of the basis of its parent.

Norm(x): AlgQuatElt -> FldElt

Norm(x): AlgQuatOrdElt -> RngElt

The reduced norm \({\rm N}(x)\) of the element \(x\) of a quaternion algebra, defined so that the characteristic polynomial for \(x\) is \(x^2 - {\operatorname{Tr}}(x)x + {\rm N}(x) = 0\), where \({\operatorname{Tr}}(x)\) is the reduced trace.

Trace(x): AlgQuatElt -> FldElt

Trace(x): AlgQuatOrdElt -> RngElt

The reduced trace \({\operatorname{Tr}}(x)\) of the element \(x\) of a quaternion algebra, defined so that the characteristic polynomial for \(x\) is \(x^2 - {\operatorname{Tr}}(x)x + {\rm N}(x) = 0\), where \({\rm N}(x)\) is the reduced norm.

CharacteristicPolynomial(x): AlgQuatElt -> RngUPolElt

CharacteristicPolynomial(x): AlgQuatOrdElt -> RngUPolElt

The characteristic polynomial of degree \(2\) for the element \(x\) of a quaternion algebra over the base ring of its parent.

MinimalPolynomial(x): AlgQuatElt -> RngUPolElt

The minimal polynomial of degree \(1\) or \(2\) for the element \(x\) of a quaternion algebra over the base ring of its parent.

Example: Element Arithmetic (ex-317c57)

We demonstrate the relation between characteristic polynomial, and reduced trace and norm in the following example.

> A := QuaternionAlgebra< RationalField() | -17, -271 >;
> x := A![1,-2,3,0];
> Trace(x);
2
> Norm(x);
2508
> x^2 - Trace(x)*x + Norm(x);
0

Run in calculatorCopy input

Note that trace and norm of an element \(x\) of any algebra can be defined as the trace and norm of the linear operator corresponding to right-multiplication by \(x\). The reduced trace and norm in a quaternion algebra \(A\) are taken instead to be the corresponding trace and determinant in any two-dimensional matrix representation of \(A\), or equivalently, the sum and product of an element with its conjugate. The definition of norm and trace used for a general algebra can be realised in a quaternion algebra by the following code.

> P<X> := PolynomialRing(RationalField());
> M := RepresentationMatrix(x, A);
> M;
[   1   -2    3    0]
[  34    1    0    3]
[-813    0    1    2]
[   0 -813  -34    1]
> Trace(M);
4
> Factorization(CharacteristicPolynomial(M));
[
    <X^2 - 2*X + 2508, 2>
]

Run in calculatorCopy input

The general definition of trace (for the algebra) is twice the reduced trace, and the general definition of norm is the square of the reduced norm.