Subsections

• Ideal Creation
• Ideal Arithmetic
• Invariants
• Basis Representation
• Predicates on Ideals
• Ideal Factorization
• Other Ideal Functions

O * x : RngFunOrdElt, RngFunOrd -> RngFunOrdIdl

Create the ideal x * O.

The order O that the ideal I is defined over.

The product IJ of the (fractional) ideals I and J, generated by the products of elements of I and elements of J.

The quotient of the (fractional) ideals I and J of an order O. This is the fractional ideal K of O with the property that J K=I.

The sum of the (fractional) ideals I and J, generated by the sums of elements of I and elements of J.

The k-th power of the (fractional) ideal I (for an integer k).

I * x : FldFunElt, RngFunOrdIdl -> RngFunOrdIdl

Given an element x of (or coercible into) a function field K, and a (fractional) ideal I in an order of K, return the product of the ideal and the principal ideal generated by x.

Create the fractional ideal I / x

Lcm(I, J) : RngFunOrdIdl, RngFunOrdIdl -> RngFunOrdIdl

LeastCommonMultiple(I, J) : RngFunOrdIdl, RngFunOrdIdl -> RngFunOrdIdl

Return the least common multiple of ideals I and J. They must both be defined over the same maximal order.

The denominator of the fractional ideal I. This is the smallest positive integer d such that dI is an integral ideal.

Returns, for a prime ideal I of an order O lying over the rational prime p, the maximal exponent e such that I^e divides the principal ideal pO. If p is not given then the minimal integer lying in I is used.

Given a prime ideal I this function returns the residue class degree f of the ideal I.

Given an ideal I of an order O of the function field K, this function returns an integral basis for I as a sequence of elements of K.

Returns the basis matrix for the ideal I of O. The basis matrix consists of the elements of an integral basis for the ideal written as rows of rational function coefficients with respect to the power basis of the function field K of which O is an order.

Returns the transformation matrix for the ideal I of O, as well as a denominator. The transformation matrix consists of the elements of an integral basis for the ideal written as columns of coefficients with respect to the basis of the order O.

The predicates listed below allow fractional ideals as argument.

I eq J : RngFunOrdIdl, RngFunOrdIdl -> BoolElt

I ne J : RngFunOrdIdl, RngFunOrdIdl -> BoolElt

x in I : FldFunElt, RngFunOrdIdl -> BoolElt

x in I : RngFunOrdElt, RngFunOrdIdl -> BoolElt

x notin I : FldFunElt, RngFunOrdIdl -> BoolElt

x notin I : RngFunOrdElt, RngFunOrdIdl -> BoolElt

True if and only if the fractional ideal I is integral.

True if I is a prime ideal; false otherwise.

True if I is the zero ideal, false otherwise.

Factorization of the ideal I (as sequence of prime ideal, exponent pairs).

Given a (fractional) ideal I of O, return a sequence of elements of K that generate I as an ideal.

Returns the multiplicator ring of the ideal I of the order O, that is, the subring of elements of the field of fractions of O that multiply I into itself.

Given a prime ideal I of an order O, and an element w in O, this function returns the valuation v_I(w) of w with respect to I, which is a non-negative integer.

Valuation(I, J) : RngFunOrdIdl, RngFunOrdIdl -> RngIntElt

Given a prime ideal I and an ideal J in an order O, return the valuation v_I(J) of J at I, that is, the number of factors I in the prime ideal decomposition of J. Note that, since the ideal J is allowed to be a fractional ideal, the returned value may be a negative integer.