Citation
Sitaraman, Sankar (1994) Arithmetic of cyclotomic fields and Fermattype equations. Dissertation (Ph.D.), California Institute of Technology. https://resolver.caltech.edu/CaltechTHESIS:05162013100413388
Abstract
Let l be any odd prime, and ζ a primitive lth root of unity. Let C_l be the lSylow subgroup of the ideal class group of Q(ζ). The Teichmüller character w : Z_l → Z^*_l is given by w(x) = x (mod l), where w(x) is a p1st root of unity, and x ∈ Z_l. Under the action of this character, C_l decomposes as a direct sum of C^((i))_l, where C^((i))_l is the eigenspace corresponding to w^i. Let the order of C^((3))_l be l^h_3). The main result of this thesis is the following: For every n ≥ max( 1, h_3 ), the equation x^(ln) + y^(ln) + z^(ln) = 0 has no integral solutions (x,y,z) with l ≠ xyz. The same result is also proven with n ≥ max(1,h_5), under the assumption that C_l^((5)) is a cyclic group of order l^h_5. Applications of the methods used to prove the above results to the second case of Fermat's last theorem and to a Fermatlike equation in four variables are given.
The proof uses a series of ideas of H.S. Vandiver ([Vl],[V2]) along with a theorem of M. Kurihara [Ku] and some consequences of the proof of lwasawa's main 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 not divide the class number h^+ of the maximal real subfield of Q(e^(2πi/i)). The crucial gap 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.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  Mathematics 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Thesis Availability:  Public (worldwide access) 
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Defense Date:  24 May 1994 
Record Number:  CaltechTHESIS:05162013100413388 
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:05162013100413388 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  7715 
Collection:  CaltechTHESIS 
Deposited By:  Benjamin Perez 
Deposited On:  16 May 2013 18:36 
Last Modified:  02 Dec 2020 02:20 
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