Computations of class numbers

of real quadratic fields

Author:
Anitha Srinivasan

Journal:
Math. Comp. **67** (1998), 1285-1308

MSC (1991):
Primary 11A51

DOI:
https://doi.org/10.1090/S0025-5718-98-00965-X

MathSciNet review:
1468944

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper an unconditional probabilistic algorithm to compute the class number of a real quadratic field is presented, which computes the class number in expected time . 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 . Previous algorithms with the above running time , obtain an approximation for by assuming an appropriate extension of the Riemann Hypothesis. Our algorithm finds an appoximation for 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 . However, our estimate of on the running time of our algorithm to compute the class number is not effective.

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Additional Information

**Anitha Srinivasan**

Affiliation:
Department of Mathematics, University of Georgia, Athens, Georgia 30602

Address at time of publication:
Department of Mathematics, University of Puerto Rico, CUH Station, 100 Carretera 908, Humacao, Puerto Rico 00791-4300

Email:
as@turing.upr.clu.edu

DOI:
https://doi.org/10.1090/S0025-5718-98-00965-X

Keywords:
Class number,
binary quadratic forms,
real quadratic field,
regulator

Received by editor(s):
July 2, 1996

Received by editor(s) in revised form:
January 31, 1997

Article copyright:
© Copyright 1998
American Mathematical Society