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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- The algorithm for computing roots of elements has been replaced
by a faster one.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
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.
Next: Quadratic Fields [HB 54]
Up: Algebraic Number Fields [HB
Previous: Fields and Orders