Algebraic Number Fields [HB 53]

Algebraic number fields and their orders have been modified in several ways since Magma V2.7. A new field type has been created to act as the field of fractions of orders of number fields. More functionality has been provided for relative fields and orders and their ideals and quotients by (relative) ideals can now be created.

Removals:

• ThueSolveInexact, BetterPolynomial, ClassGroupStructure and IsRelative have all been removed as they were considered to be obsolete. The functionality of each of these can be achieved by alternate means.

• MinimalInteger has been removed but its functionality can be achieved by using the function meet with second argument the coefficient ring of the order the ideal is of. There are more possibilities for the second argument to this function.

• Polynomials can no longer be coerced into number fields or orders. This removes the ambiguity when coercing polynomials into a ring with a number field or an order as the coefficient ring.

• NumberField of cyclotomic and quadratic fields was withdrawn.

• SetKantVerbose has been removed but verbose printing can be gained using SetVerbose.

General New Features:

• Predicates on orders, fields and ideals: IsAbsoluteOrder, IsAbsoluteField, IsAlgebraicField, IsNumberField, IsSimple, IsRamified, IsUnramified, IsTotallyRamified, IsWildlyRamified, IsTamelyRamified, IsInert, IsSplit, IsTotallySplit.