Explicit Solutions of Imaginary Quadratic Norm Equations.

摘要

Let K = Q(√-d) be an imaginary quadratic extension of Q. Let h be the class number and D be the discriminant of the field K. Assume p is a prime such that D/ p = 1. Then p splits in K. The elements of the ring of integers OK are of the form x + √-d y if d ≡ 1,2 (mod 4) and x + (√- d/2)y if d ≡ 3, where x and y ∈ Z. The norm N K/Q(x+√- dy) = x2 + dy 2 and NK/ Q(x + (√-d/2) y) = (2x + y)2/4 + dy2/4. In this thesis, we find the elements of norm ph explicitly. We also prove certain congruences for solutions of norm equations.