Let l be any odd prime, and ζ a primitive l-th root of unity. Let C_l be thel-Sylow subgroup of the ideal class group of Q(ζ). The Teichmüller characterw : Z_l → Z^*_l is given by w(x) = x (mod l), where w(x) is a p-1-st root of unity,and x ∈ Z_l. Under the action of this character, C_l decomposes as a direct sum ofC^((i))_l, where C^((i))_l is the eigenspace corresponding to w^i. Let the order of C^((3))_l bel^h_3). The main result of this thesis is the following: For every n ≥ max( 1, h_3 ), theequation x^(ln) + y^(ln) + z^(ln) = 0 has no integral solutions (x,y,z) with l ≠ xyz. Thesame result is also proven with n ≥ max(1,h_5), under the assumption that C_l^((5)) isa cyclic group of order l^h_5. Applications of the methods used to prove the aboveresults to the second case of Fermat's last theorem and to a Fermat-like equationin four variables are given.
The proof uses a series of ideas of H.S. Vandiver ([Vl],[V2]) along with atheorem of M. Kurihara [Ku] and some consequences of the proof of lwasawa'smain conjecture for cyclotomic fields by B. Mazur and A. Wiles [MW]. In [V1]Vandiver claimed that the first case of Fermat's Last Theorem held for l if l did notdivide the class number h^+ of the maximal real subfield of Q(e^(2πi/i)). The crucialgap in Vandiver's attempted proof that has been known to experts is explained,and complete proofs of all the results used from his papers are given.
