Introduction

Quadratic fields in Magma can be created as a subtype of the number fields FldNum. The advantage of the special quadratic fields is that some special (faster) algorithms have been or will be implemented to deal with them; the functions for the special quadratic fields (created with the QuadraticField function) are described here. Functions which work generally for number fields and their orders are described in Chapter ORDERS AND ALGEBRAIC FIELDS.

The categories involved are FldQuad, for fields, and RngQuad for their orders.

Subsections

• Representation

Representation

For every squarefree integer d (not 0 or 1) there is a unique quadratic field Q(Sqrt(d)); for any integer k we have the field Q(Sqrt(k^2d)) isomorphic to Q(Sqrt(d)). Given any integer m, the function QuadraticField will create a structure corresponding to the quadratic field Q(Sqrt(d)), where d is the squarefree kernel of m (d will have the same sign as m and its absolute value is the largest squarefree divisor of m). In Magma a list of quadratic fields currently present is maintained, and if Q(Sqrt(d)) has been created before a reference on it will be returned: two fields with the same d are the same. The discriminant D of Q(Sqrt(d)) will be D=d if d equiv1bmod4 and D=4d if d equiv2, 3bmod4.

Elements of Q(Sqrt(d)) are represented by a common positive denominator b and two integer coefficients: alpha=(1/b)(x + ySqrt(d)).

The ring of integers of F=Q(Sqrt(d)) will be O_F=Z + epsilon_dZ, where epsilon_d=cases( Sqrt(d)&if d = 2, 3 mod 4, cr (1 + Sqrt(d)/2)&if d = 1 mod 4.cr) Elements of O_F are represented by two integer coefficients alpha=x + yepsilon_d. The pair 1, epsilon_d forms an integral basis for F=Q(Sqrt(d)), but note that elements of F are represented using the basis of the equation order (1, Sqrt(d)) instead.

For any positive integer f there is a suborder of conductor f in O_F, whose elements are of the form x + y f epsilon_d, for any integers x, y. The discriminant of the order of conductor f is f^2D, where D is the field discriminant.

The equation order of F is E_F = Z + Sqrt(d)Z. Suborders of conductor f can be formed which will contain elements of the form x + y f Sqrt(d) for any integers x and y.