  The absolute Galois group of the cyclotomic field $K={\mathbb Q}(\zeta_p)$
acts on the \'etale homology of the Fermat curve $X$ of exponent $p$. We study
a Galois cohomology group which is valuable for measuring an obstruction for
$K$-rational points on $X$. We analyze a $2$-nilpotent extension of $K$ which
contains the information needed for measuring this obstruction. We determine a
large subquotient of this Galois cohomology group which arises from Heisenberg
extensions of $K$. For $p=3$, we perform a Magma computation with ray class
fields, group cohomology, and Galois cohomology, which determines it
completely.
