Publication Date

Spring 2026

Degree Type

Thesis

Degree Name

Master of Science (MS)

Department

Mathematics and Statistics

Advisor

Jordan Schettler; Tim Hsu; Yan Zhang

Abstract

Fermat’s Last Theorem states that the equation xn + yn = zn has no nontrivial integer solutions for n ≥ 3. While the rational case is completely solved, the situation over quadratic number fields is not as straightforward. For some exponents nontrivial solutions exist, while for others they do not. This thesis studies Fermat-type equations over quadratic number fields, combining classical methods with more modern techniques from the theory of elliptic curves. The thesis first reviews the rational cases n = 2, n = 3, n = 4, then develops the necessary background on quadratic number fields, including rings of integers, algebraic integers, norms, traces, class numbers, and unique factorization. It then turns to equations over quadratic fields, beginning with n = 3, where Burnside’s parametrization gives nontrivial solutions in certain quadratic fields. Building on this, the thesis proves that the equations x6 + y6 = z6 and x9 + y9 = z9 have no nontrivial solutions in any quadratic number field by expounding on Aigner’s work. These arguments are then extended to exponents of the form 3m, including a completion of a gap in the recent argument of Tho. The final section discusses the special field Q(√2), where Jarvis and Meekin proved Fermat’s equation has no nontrivial solution in Q(√2) for every exponent n ≥ 4 using methods similar to the ones used by Ribet and Wiles to prove Fermat’s Last Theorem for integers.

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