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Subsections
Zero(A) : AlgQuat -> AlgQuatElt
The zero element of the quaternion algebra A.
One(A) : AlgQuat -> AlgQuatElt
The identity element of the quaternion algebra A.
S . i : AlgQuat, RngIntElt -> AlgQuatElt
Given a quaternion algebra A or order S and an integer 1 <= i <= 3,
returns the i-th generator 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.
The sum of x and y.
The difference of x and y.
The product of x and y.
The quotient of x by the unit y in the quaternion algebra.
Returns true if the elements a and b of an algebra A 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 -> AlgQuatElt
The conjugate bar(x) of x, defined such that the reduced trace
and reduced norm are bar(x) + x and bar(x)x, respectively.
ElementToSequence(x) : AlgQuatElt -> SeqEnum
Eltseq(x) : AlgQuatElt -> SeqEnum
Eltseq(x) : AlgQuatOrdElt -> SeqEnum
Coordinates(x) : AlgQuatElt -> SeqEnum
Coordinates(x) : AlgQuatOrdElt -> SeqEnum
Given an element x of a quaternion algebra, order, or ideal, 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 x, defined such that the
characteristic polynomial for x is x^2 - Tr(x)x + N(x) = 0,
where Tr(x) is the reduced trace.
Trace(x) : AlgQuatOrdElt -> RngElt
The reduced trace Tr(x) of x, defined such that the characteristic
polynomial for x is x^2 - Tr(x)x + N(x) = 0, where N(x) is the
reduced norm.
CharacteristicPolynomial(x) : AlgQuatOrdElt -> RngUPolElt
The characteristic polynomial of degree 2 for x over the base ring
of its parent.
MinimalPolynomial(x) : AlgQuatOrdElt -> RngUPolElt
The minimal polynomial of degree 1 or 2 for x 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);
5
> Norm(x);
1640
> 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 of norm of the linear operator of left or right
multiplication by x. The reduced trace and norm in a quaternion
algebra A are instead the corresponding trace or norm in any two
dimensional matrix representation of A, generally over an extension
of the base field. The former definition for a general algebra can
be realised with the following lines.
> P<X> := PolynomialRing(RationalField());
> M := Matrix(4,4,&cat[ Eltseq(x*y) : y in Basis(A) ]);
> Trace(M);
10
> Factorization(CharacteristicPolynomial(M));
[
<X^2 - 5*X + 1640, 2>
]
And it is easily verified that the general definition of trace is twice
the reduced trace, and the general definition of norm is the square of
the reduced norm.
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