The asymptotic Fermat conjecture states that for a number field K there is a constant B_K such that for primes p≥B_K the only K-rational points on the Fermat curve X^p+Y^p+Z^p=0 are the obvious ones. In this talk we survey joint work with Nuno Freitas, Alain Kraus and Haluk Sengun, on the asymptotic Fermat conjecture. In particular we prove AFC for family of number fields K=ℚ(ζ_2^r)^+.
