Jove
Visualize
Contact Us

Related Experiment Videos

A Simple Proof of Siegel's Theorem.

D M Goldfeld1

  • 1The Institute for Advanced Study, Princeton, New Jersey 08540.

Proceedings of the National Academy of Sciences of the United States of America
|April 1, 1974
PubMed
Summary
This summary is machine-generated.

Related Concept Videos

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same journal

Chemotactic self-organization captures the dynamics of mammalian hair follicle patterning.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Tomographic imaging of superconducting order using particle-hole interference.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Inhibitory potential of autologous neutralizing antibodies sets quantitative limits on the rebound-competent HIV-1 reservoir.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Inferring epidemiological parameters under an infectious phylogeography model with visitor dynamics.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Analytical modeling for suction cup designs for skin-interfaced wearable devices.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Improving cell-free metabolism through direct integration of artificial respiratory chains.

Proceedings of the National Academy of Sciences of the United States of America·2026
See all related articles
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

This study presents a simple proof for Siegel's theorem on class numbers, showing that the class number h(d) of quadratic fields grows faster than d to the power of 1/2. This advances understanding in number theory without complex algebraic methods.

Area of Science:

  • Number Theory
  • Algebraic Number Theory
  • Quadratic Fields

Background:

  • Siegel's theorem provides a lower bound for the class number of quadratic fields.
  • Previous proofs often relied on advanced algebraic number theory.
  • Class number, h(d), is a fundamental invariant of quadratic fields Q(sqrt(d)).

Purpose of the Study:

  • To present a brief and simple proof of Siegel's theorem.
  • To demonstrate a proof not requiring algebraic number theory.
  • To provide accessible insights into the growth of class numbers.

Main Methods:

  • A novel, simplified proof methodology.
  • Avoidance of complex algebraic number theory concepts.
  • Direct mathematical derivation based on fundamental principles.

Related Experiment Videos

Main Results:

  • A concise proof of the inequality h(d) >> d(1/2) as d approaches infinity.
  • Demonstration that the class number grows significantly faster than the square root of d.
  • Confirmation of Siegel's celebrated theorem through elementary methods.

Conclusions:

  • The study successfully provides a simple proof for Siegel's theorem.
  • This proof is accessible to a broader mathematical audience.
  • It highlights that advanced algebraic machinery is not always necessary for fundamental number theoretic results.