Topology of Numbers · Problem 8.3.4

Quadratic Fields — Allen Hatcher

Mathematics — Number Theory · quadratic-fields-ideals-lattices

Problem

(a) For a form

ax^2 + bxy + cy^2

of discriminant

\Delta

we have the associated ideal

L(a, \frac{b+\sqrt{\Delta}}{2})

whose basis

a, \frac{b+\sqrt{\Delta}}{2}

determines a parallelogram

P

. When

\Delta < 0

show that

P

is a rhombus if and only if

a = c

. (b) Give an example of a form

ax^2 + bxy + ay^2

with

\Delta > 0

for which

P

is not a rhombus.