Algebraic Number Fields [HB 50]

Changes:

• The ideal<> constructor now checks that the input defines an ideal.

• Factorization of ideals has been improved by storing a product representation on appropriate ideals and using a coprime factorization algorithm on this partial factorization.

• Changed SplittingField for polynomials over Q to allow it to return a tower of fields rather than a simple optimized representation.

• Improved performance of OptimizedRepresentation and LLL by using a different LLL-version. Furthermore, if this function fails to find a better representation, the old representation is returned.

• A rewrite of the code computing completions of number fields and prime ideals using the new local rings and allowing for precision change.

New Features:

• Homomorphisms from orders of algebraic number fields can be created by giving the images of the basis elements (not just the primitive element).

• The greatest common divisor of two ideals can be obtained.

Bug Fixes:

• Bug fix in IsSubfield, in the pre-image code for homomorphisms between orders and number fields, in the pre-image code for the unit-map.