To utilize the force of Kummer’s main theorem, it is imperative to determine when a prime p is regular. In theory, at least, this is simple. All one needs to do is to compute the class number of the cyclotomic field K p = ℚ(ζ p ), where ζ p is a primitive pth root of 1. And so Kummer began by discovering formulas for the class number. However, it soon became apparent that the computations involved were much too difficult, and other more suitable criteria for regularity were needed. These were also discovered by Kummer.


Prime Ideal Class Number Bernoulli Number Arithmetic Property Principal Character 
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Copyright information

© Springer-Verlag New York Inc. 1979

Authors and Affiliations

  • Paulo Ribenboim
    • 1
  1. 1.Department of Mathematics and StatisticsJeffery Hall Queen’s UniversityKingstonCanada

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