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For more information about elements of orders of associative algebras,
see Section Orders.
Zero(A) : AlgQuat -> AlgQuatElt
The zero element of the quaternion algebra A.
One(A) : AlgQuat -> AlgQuatElt
The identity element of the quaternion algebra A.
Name(A, i) : AlgQuat, RngIntElt -> AlgQuatElt
Given a quaternion algebra A and an integer 1≤i≤3,
returns the ith 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.
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.
The sum of x and y.
The difference of x and y.
The product of x and y.
x / y : AlgQuatOrdElt, AlgQuatOrdElt -> AlgQuatElt
The quotient of x by the unit y in the quaternion algebra.
Returns true if the elements x and y are equal; otherwise false.
Returns true if and only if the elements x and y are not equal.
Returns true if and only if x is in the algebra A.
Returns true if and only if x is not in the algebra A.
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.
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) : AlgQuatOrdElt -> RngElt
The reduced norm N(x) of the element x of a quaternion algebra,
defined so that the characteristic polynomial for x is
x2 - Tr(x)x + N(x) = 0, where Tr(x) is the reduced trace.
Trace(x) : AlgQuatOrdElt -> RngElt
The reduced trace Tr(x) of the element x of a quaternion algebra,
defined so that the characteristic polynomial for x is
x2 - Tr(x)x + N(x) = 0, where N(x) is the reduced norm.
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.
The minimal polynomial of degree 1 or 2 for the element x of a quaternion
algebra over the base ring of its parent.
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
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>
]
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.
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