Quadratic Fields [HB 54]

The quadratic fields and rings have been rewritten to become a part of the algebraic fields and their orders. Quadratic Fields (FldQuad) now inherit from the Number Fields (FldNum) and Quadratic Rings (RngQuad) now inherit from the Orders of algebraic fields (RngOrd).

Removals and Changes:

• Some element functions available for a range of discriminants only are now available for one less discriminant (17), (div, mod, Modexp, Gcd). These functions along with Factorization and TrialDivision are only for elements in maximal orders.

• The field of fractions of a quadratic ring is a field of type FldOrd. To retrieve the corresponding quadratic field the function NumberField must be used.

• Elements are represented with respect to the basis of the order or field which is their parent. For example, EO.2, where EO is an equation order of some field where it is different to the maximal order of the field, is $\sqrt{d}$ and EO has basis $\{ 1, \sqrt{d}\}$. Previously, the elements of such an order were expressed using the basis of the maximal order $\{ 1, (1 + \sqrt{d})/2\}$.

• O.1 where O is a quadratic order now returns an element in the field of fractions of O which will be the first basis element of O instead of the second. O.2 will return the second basis element which was before returned by O.1. Name will also return the 2nd basis element of O and AssignNames will assign the string to this 2nd basis element.

• The algorithm used for ClassGroup and ClassNumber has changed for small discriminants. The defaults of the parameters have changed for large discriminants resulting in a longer running time and results that are provable under GRH. ``ClassGroup'' for non maximal orders has been renamed to PicardGroup and its size to PicardNumber.

• Taking the ith coordinate of an element of a quadratic order x[i] returns a rational instead of an integer.

• The defining polynomial of any order is the same as that of its Quadratic field; it does not reflect the presence of the conductor.

• Regulator returns a FldPrElt instead of a FldReElt.

• NormEquation returns a sequence containing possibly more than one element as its second return value.

• BiquadraticResidueSymbol and Primary take arguments of gaussian integer (quadratic ring elements) instead of field elements since an error occurred when field elements were input.

New Features:

• Quadratic fields and their orders are compatible with number fields and their orders.

• Quadratic fields and orders can be created from number fields and their orders using the function IsQuadratic.

• All the functionality of the orders and algebraic fields which was absent for the quadratic fields is now present. Some examples are SplittingField, PrimitiveElement, AutomorphismGroup and GaloisGroup.

• NormEquation is now possible for real quadratic fields.

• A quadratic field may be extended to a relative number field.

• Ideals (both integral and fractional) of quadratic orders may be formed. In addition to the functions for ideals of orders the functions Conjugate, Content, Discriminant, QuadraticForm and Reduction are available for quadratic ideals. Quadratic ideals can also be created from a quadratic form using the function QuadraticIdeal.

• Quotients of quadratic orders by ideals can be taken. The result will have type RngOrdRes.

• A LCM function has been added for elements of a maximal order.

Bug Fixes:

• A bug in FundamentalUnit of a quadratic order has been fixed by the overall changes.

• A bug which resulted in incorrect answers being produced by BiquadraticResidueSymbol has been corrected. An example which did not finish running in a reasonable amount of time now does.

• A bug which caused GCD computations to crash has been fixed.