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The absolute Galois group of the cyclotomic field K = Q (ζ p) acts on the étale 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. [ABSTRACT FROM AUTHOR]

For an odd prime p satisfying Vandiver's conjecture, we give explicit formulae for the action of the absolute Galois group G Q ( ζ p ) on the homology of the degree p Fermat curve, building on work of Anderson. Further, we study the invariants and the first Galois cohomology group which are associated with obstructions to rational points on the Fermat curve. [ABSTRACT FROM AUTHOR]