Elements, Ideals and Quotients

Changes:

• The algorithm for computing roots of elements has been replaced by a faster one.

• The i-th row of RepresentationMatrix of an element now gives the coefficients from the multiplication of an element by the i-th basis element instead of the i-th column.

• Eltseq for an order element will always return field elements. (This is because in general relative extensions the coefficients have to be non-integral).

• ideal<O | mat> now needs a transposed matrix, i.e. the rows of mat correspond to the ideal basis, not the columns.

• The codomain of the map returned by SUnitGroup is now the field rather than the order.

New features:

• Fractional ideals of orders have been given their own type, RngOrdFracIdl. All ideals inherit from this type.

• The functionality for absolute ideals has been provided for relative ideals where possible.

• Quotients of a relative order and an ideal of that order can now be formed. Ideals can be constructed from a module over an order or a matrix and ideals of the coefficient order. This information is used for the basis of the ideal.

• The functions IsPower, Root, IsSquare and SquareRoot have been implemented for ideals of orders.

• The Norm, Trace, CharacteristicPolynomial, MinimalPolynomial and RepresentationMatrix of an element can be found in or over a user specified ring.

• Divisors is now available for elements of maximal orders.

• The index of the module $\mbox{\bf Z}[a]$ in the order containing the element a can be calculated using Index.

• UnitGroup for absolute maximal orders modulo integral ideals. RayResidueRings i.e. UnitGroups of orders modulo integral ideals and restriction of the signs of the embeddings. Also provided is ChineseRemainderTheorem for infinite places.

• pRadical, MultiplicatorRing for non primes and relative extensions.

• ColonIdeal, AbsoluteNorm for ideals, id meet r.

• Inverses and division in residue class rings.

• A special algorithm for PowerProduct using matrix input.

• !! on ideals will create the ideal as an ideal of the ring given.

• Elements can be indexed by integers, for example, x[i], to return the ith coefficient of an element x. The coefficients of an element with respect to a Q-basis can be returned using Flat.

• Ramification theory for ideals: RamificationGroup, RamificationField, DecompositionGroup, DecompositionField, InertiaGroup, InertiaField.

• subset for ideals. Automorphisms of number fields may be applied to ideals.

Bug fixes:

• All functions returning real conjugates of algebraic numbers now do a precision check, thus the returned values are correct.

• Certain class group computations that failed because of overflow in the real computations work now.

• The move system is more robust wrt. circular references.

• Coercing a polynomial into a ring with an order or number field as the coefficient ring no longer sometimes coerces the polynomial into the coefficient ring but always over the coefficient ring.

• Repeated principal ideal testing without the computation of the generator produced wrong results.