Citation
Davis, Daniel Lee (1969) On the Distribution of the Signs of the Conjugates of the Cyclotomic Units in the Maximal Real Subfield of the qth Cyclotomic Field, q a Prime. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/YG5GHE68. https://resolver.caltech.edu/CaltechTHESIS:02012016080841282
Abstract
Let F = Ǫ(ζ + ζ^{ –1}) be the maximal real subfield of the cyclotomic field Ǫ(ζ) where ζ is a primitive qth root of unity and q is an odd rational prime. The numbers u_{1}=1, u_{k}=(ζ^{k}ζ^{k})/(ζζ^{1}), k=2,…,p, p=(q1)/2, are units in F and are called the cyclotomic units. In this thesis the sign distribution of the conjugates in F of the cyclotomic units is studied.
Let G(F/Ǫ) denote the Galoi's group of F over Ǫ, and let V denote the units in F. For each σϵ G(F/Ǫ) and μϵV define a mapping sgn_{σ}: V→GF(2) by sgn_{σ}(μ) = 1 iff σ(μ) ˂ 0 and sgn_{σ}(μ) = 0 iff σ(μ) ˃ 0. Let {σ_{1}, ... , σ_{p}} be a fixed ordering of G(F/Ǫ). The matrix M_{q}=(sgn_{σj}(v_{i}) ) , i, j = 1, ... , p is called the matrix of cyclotomic signatures. The rank of this matrix determines the sign distribution of the conjugates of the cyclotomic units. The matrix of cyclotomic signatures is associated with an ideal in the ring GF(2) [x] / (x^{p}+ 1) in such a way that the rank of the matrix equals the GF(2)dimension of the ideal. It is shown that if p = (q1)/ 2 is a prime and if 2 is a primitive root mod p, then M_{q} is nonsingular. Also let p be arbitrary, let ℓ be a primitive root mod q and let L = {i  0 ≤ i ≤ p1, the least positive residue of defined by ℓ^{i} mod q is greater than p}. Let H_{q}(x) ϵ GF(2)[x] be defined by H_{q}(x) = g. c. d. ((Σ x^{i}/I ϵ L) (x+1) + 1, x^{p} + 1). It is shown that the rank of M_{q} equals the difference p  degree H_{q}(x).
Further results are obtained by using the reciprocity theorem of class field theory. The reciprocity maps for a certain abelian extension of F and for the infinite primes in F are associated with the signs of conjugates. The product formula for the reciprocity maps is used to associate the signs of conjugates with the reciprocity maps at the primes which lie above (2). The case when (2) is a prime in F is studied in detail. Let T denote the group of totally positive units in F. Let U be the group generated by the cyclotomic units. Assume that (2) is a prime in F and that p is odd. Let F_{(2)} denote the completion of F at (2) and let V_{(2)} denote the units in F_{(2)}. The following statements are shown to be equivalent. 1) The matrix of cyclotomic signatures is nonsingular. 2) U∩T = U^{2}. 3) U∩F^{2}_{(2)} = U^{2}. 4) V_{(2)}/ V_{(2)}^{2} = ˂v_{1} V_{(2)}^{2}˃ ʘ…ʘ˂v_{p} V_{(2)}^{2}˃ ʘ ˂3V_{(2)}^{2}˃.
The rank of M_{q} was computed for 5≤q≤929 and the results appear in tables. On the basis of these results and additional calculations the following conjecture is made: If q and p = (q 1)/ 2 are both primes, then M_{q} is nonsingular.
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)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  4 April 1969  
Funders: 
 
Record Number:  CaltechTHESIS:02012016080841282  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:02012016080841282  
DOI:  10.7907/YG5GHE68  
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  9554  
Collection:  CaltechTHESIS  
Deposited By:  INVALID USER  
Deposited On:  02 Feb 2016 16:39  
Last Modified:  26 Apr 2024 21:01 
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