Computations of class numbers of real quadratic fields

In this paper an unconditional probabilistic algorithm to compute the class number of a real quadratic field Q( √ d) is presented, which computes the class number in expected time O(d1/5+ ). The algorithm is a random version of Shanks’ algorithm. One of the main steps in algorithms to compute the class number is the approximation of L(1, χ). Previous algorithms with the above running time O(d1/5+ ), obtain an approximation for L(1, χ) by assuming an appropriate extension of the Riemann Hypothesis. Our algorithm finds an appoximation for L(1, χ) without assuming the Riemann Hypothesis, by using a new technique that we call the ‘Random Summation Technique’. As a result, we are able to compute the regulator deterministically in expected time O(d1/5+ ). However, our estimate of O(d1/5+ ) on the running time of our algorithm to compute the class number is not effective.

• Type: Journal
• Year: 1998
• Where: MOC
• Authors: Anitha Srinivasan
• Keywords: Algorithm, Class Number, MOC 1998, Unconditional Probabilistic Algorithm