Elements of Orders

Construction of Elements

O ! 0: AlgAssVOrd, RngIntElt -> AlgAssVOrdElt

Zero(O): AlgAssVOrd -> AlgAssVOrdElt

The zero element of the associative order \(O\).

O ! 1: AlgAssVOrd, RngIntElt -> AlgAssVOrdElt

One(O): AlgAssVOrd -> AlgAssVOrdElt

The identity element of the associative order \(O\).

O . i: AlgAssVOrd, RngIntElt -> AlgAssVElt

Given an associative order \(O\) and an integer \(i\), returns the \(i\)th basis element as an order over the base ring. Note that the element \(1\) may or may not be the first element of a basis. These basis elements are returned as elements of the algebra of \(O\) not as elements of \(O\) itself.

O ! x: AlgAssVOrd, Any -> AlgAssVOrdElt

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

Arithmetic of Elements

x + y: AlgAssVOrdElt, AlgAssVOrdElt -> AlgAssVOrdElt

The sum of elements \(x\) and \(y\) of an order of an associative algebra.

x - y: AlgAssVOrdElt, AlgAssVOrdElt -> AlgAssVOrdElt

The difference of elements \(x\) and \(y\) of an order of an associative algebra.

- x: AlgAssVOrdElt -> AlgAssVOrdElt

The negation of element \(x\) of an order of an associative algebra.

x * y: AlgAssVOrdElt, AlgAssVOrdElt -> AlgAssVOrdElt

The product of elements \(x\) and \(y\) of an associative algebra.

u * c: AlgAssVOrdElt, RngElt -> AlgAssVOrdElt

c * u: RngElt, AlgAssVOrdElt -> AlgAssVOrdElt

The product of the element \(u\) of an associative order by the scalar \(c\).

x / y: AlgAssVOrdElt, AlgAssVOrdElt -> AlgAssVElt

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

x div y: AlgAssVOrdElt, AlgAssVOrdElt -> AlgAssVOrdElt

The exact division of \(x\) by \(y\) in the order containing them.

x ^ n: AlgAssVOrdElt, RngIntElt -> AlgAssVOrdElt

The product of the element \(x\) of an associative order with itself \(n\) times.

Predicates on Elements

x eq y: AlgAssVOrdElt, AlgAssVOrdElt -> BoolElt

Returns true if and only if the elements \(x\) and \(y\) are equal.

x ne y: AlgAssVOrdElt, AlgAssVOrdElt -> BoolElt

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

IsZero(x): AlgAssVOrdElt -> BoolElt

Return true if the element \(x\) of an associative order is the zero element.

IsUnit(a): AlgAssVOrdElt -> BoolElt

Return true if the element \(x\) of an associative order is a unit in that order.

IsScalar(x): AlgAssVOrdElt -> BoolElt, RngElt

Returns true if and only if \(x\) is an element of the base ring of the order containing it, and if so returns the coerced element.

Other Operations with Elements

ElementToSequence(x): AlgAssVOrdElt -> SeqEnum

Eltseq(x): AlgAssVOrdElt -> SeqEnum

Given an element \(x\) of an associative order \(O\), returns the sequence of coordinates of \(x\) in terms of the basis of \(O\).

Norm(x): AlgAssVOrdElt -> RngElt

The norm of the element \(x\) of an order as an element of its parent algebra.

Trace(x): AlgAssVOrdElt -> RngElt

The trace of the element \(x\) of an order as an element of its parent algebra.

LeftRepresentationMatrix(e): AlgAssVOrdElt -> AlgMatElt

RightRepresentationMatrix(e): AlgAssVOrdElt -> AlgMatElt

The representation matrix describing left (right) multiplication by the element \(e\) of an associative order.

RepresentationMatrix(a): AlgAssVOrdElt -> AlgMatElt

Side: MonStgElt                    Default: "Left"

The representation matrix of the element \(a\) of an associative order. This describes left multiplication unless the parameter Side is set to "Right".

CharacteristicPolynomial(x): AlgAssVOrdElt -> RngUPolElt

The characteristic polynomial of the element \(x\) of an order as an element of its parent algebra.

MinimalPolynomial(x): AlgAssVOrdElt -> RngUPolElt

The minimal polynomial of the element \(x\) of an order as an element of its parent algebra.