  Recent results of Freitas, Kraus, Sengun and Siksek, give sufficient criteria
for the asymptotic Fermat's Last Theorem to hold over a specific number field.
Those works in turn build on many deep theorems in arithmetic geometry. In this
paper we combine the aforementioned results with techniques from class field
theory, the theory of p-groups and p-extensions, Diophantine approximation and
linear forms in logarithms, to establish the asymptotic Fermat's Last Theorem
for many infinite families of number fields, and for thousands of number fields
of small degree. For example, we prove the effective asymptotic Fermat's Last
Theorem for the infinite family of fields $\mathbb{Q}(\zeta_{2^r})^+$.
