Jove
Visualize
Contact Us
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

Related Experiment Videos

On Fermat's equation over some quadratic imaginary number fields.

George C Ţurcaş1

  • 1Mathematics Institute, University of Warwick, Coventry, CV4 7AL UK.

Research in Number Theory
|April 9, 2019
PubMed
Summary

This study proves Fermat's Last Theorem over number fields using a standard conjecture from the Langlands program. It also shows no non-trivial solutions exist for Fermat's equation with prime exponents over these fields.

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

Wild Galois representations: elliptic curves with wild cyclic reduction.

Research in number theory·2026
Same journal

Endomorphism algebras of abelian varieties with large cyclic 2-torsion field over a given field.

Research in number theory·2026
Same journal

Applications of representation theory and of explicit units to Leopoldt's conjecture.

Research in number theory·2026
Same journal

Adjoint <i>L</i>-functions, congruence ideals, and Selmer groups over <math><msub><mtext>GL</mtext> <mi>n</mi></msub></math>.

Research in number theory·2026
Same journalSame Topic

Moments of derivatives of quadratic Dirichlet <i>L</i>-functions with prime conductor.

Research in number theory·2026
Same journal

Three-torsion subgroups and wild conductor exponents of plane quartics.

Research in number theory·2025

Area of Science:

  • Number Theory
  • Algebraic Geometry
  • Langlands Program

Background:

  • Fermat's Last Theorem is a famous unsolved problem in number theory.
  • The Langlands program proposes deep connections between number theory and other areas of mathematics.
  • Previous attempts to solve Fermat's Last Theorem have relied on various advanced mathematical tools.

Purpose of the Study:

  • To prove Fermat's Last Theorem over specific number fields.
  • To extend the proof to cover Fermat's equation with all prime exponents.
  • To leverage a standard conjecture within the Langlands program for these proofs.

Main Methods:

  • Utilizing a deep, standard conjecture from the Langlands program.
  • Applying advanced techniques in algebraic number theory.
Keywords:
BianchiFermatGalois representationSerre modularity

Related Experiment Videos

  • Developing novel methods to handle Fermat's equation for prime exponents.

    • Main Results:

    • A rigorous proof of Fermat's Last Theorem over the specified number fields is established.
    • It is demonstrated that Fermat's equation has no non-trivial solutions for any prime exponent over and .
    • The results confirm the power and applicability of the assumed Langlands conjecture.

    Conclusions:

    • The study provides a significant advancement in number theory by resolving Fermat's Last Theorem under a key conjecture.
    • The findings have broad implications for the Langlands program and related mathematical fields.
    • This work opens new avenues for research into Diophantine equations and their connections to automorphic forms.