Irregularity of prime numbers over real quadratic fields

  • Joshua Holden
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1423)


The concept of regular and irregular primes has played an important role in number theory at least since the time of Kummer. We extend this concept to the setting of arbitrary totally real number fields k o, using the values of the zeta function ζk0 at negative integers as our “higher Bernoulli numbers”. Once we have defined k 0-regular primes and the index of k 0-irregularity, we discuss how to compute these indices when k 0 is a real quadratic field. Finally, we present the results of some preliminary computations, and show that the frequency of various indices seems to agree with those predicted by a heuristic argument.


Zeta Function Common Denominator Class Number Negative Integer Kronecker Symbol 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Joshua Holden
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of Massachusetts at AmherstAmherstUSA

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