Let p be an odd prime. Consider the degree p Fermat curve X: x^p+y^p=z^p over K=Q(ζ_p). Let π be the étale fundamental group of X. Ellenberg and Wickelgren defined and studied certain obstructions to points on X(K) that depend on the lower central series of π. Anderson gave a formula that gives the Galois action on the homology of X and showed that it factors through a field L over K. For the next obstruction, the Galois action factors through E, a certain elementary abelian p-group extension of L. We compute a presentation for Gal(E/K) when p=3. We are unable to directly compute rk_p Gal(E/L) for p ≥ 5, although local class field theory gives some information. This poster is based on joint work with Pries and a joint project with Pries, Stojanoska, and Wickelgren.