Fermat’s last theorem over some small real quadratic fields

Title data

Freitas, Nuno ; Siksek, Samir:
Fermat’s last theorem over some small real quadratic fields.
In: Algebra & Number Theory. Vol. 9 (2015) Issue 4 . - pp. 875-895.
ISSN 1937-0652
DOI: https://doi.org/10.2140/ant.2015.9.875

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Abstract in another language

Using modularity, level lowering, and explicit computations with Hilbert modular forms, Galois representations, and ray class groups, we show that for 3 ≤ d ≤ 23, where d ≠ 5, 17 and is squarefree, the Fermat equation x^n + y^n = z^n has no nontrivial solutions over the quadratic field ℚ(d) for n ≥ 4. Furthermore, we show that for d = 17, the same holds for prime exponents n ≡ 3,5 (mod 8).