Vol. 9, No. 4, 2015

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Fermat's last theorem over some small real quadratic fields

Nuno Freitas and Samir Siksek

Vol. 9 (2015), No. 4, 875–895
Abstract

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 d5,17 and is squarefree, the Fermat equation xn + yn = zn has no nontrivial solutions over the quadratic field (d) for n 4. Furthermore, we show that for d = 17, the same holds for prime exponents n 3,5(mod8).

Keywords
Fermat, modularity, Galois representation, level lowering
Mathematical Subject Classification 2010
Primary: 11D41
Secondary: 11F80, 11F03
Milestones
Received: 16 July 2014
Revised: 23 September 2014
Accepted: 9 March 2015
Published: 30 May 2015
Authors
Nuno Freitas
Mathematisches Institut
Universität Bayreuth
95440 Bayreuth
Germany
Samir Siksek
Mathematics Institute
University of Warwick
Coventry
CV4 7AL
United Kingdom