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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


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/YG5G-HE68.


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 u1=-1, uk=(ζk-k)/(ζ-ζ-1), k=2,…,p, p=(q-1)/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 sgnσ: V→GF(2) by sgnσ(μ) = 1 iff σ(μ) ˂ 0 and sgnσ(μ) = 0 iff σ(μ) ˃ 0. Let {σ1, ... , σp} be a fixed ordering of G(F/Ǫ). The matrix Mq=(sgnσj(vi) ) , 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] / (xp+ 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 = (q-1)/ 2 is a prime and if 2 is a primitive root mod p, then Mq is non-singular. Also let p be arbitrary, let ℓ be a primitive root mod q and let L = {i | 0 ≤ i ≤ p-1, the least positive residue of defined by ℓi mod q is greater than p}. Let Hq(x) ϵ GF(2)[x] be defined by Hq(x) = g. c. d. ((Σ xi/I ϵ L) (x+1) + 1, xp + 1). It is shown that the rank of Mq equals the difference p - degree Hq(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 non-singular. 2) U∩T = U2. 3) U∩F2(2) = U2. 4) V(2)/ V(2)2 = ˂v1 V(2)2˃ ʘ…ʘ˂vp V(2)2˃ ʘ ˂3V(2)2˃.

The rank of Mq 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 Mq is non-singular.

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):
  • Taussky-Todd, Olga
Thesis Committee:
  • Unknown, Unknown
Defense Date:4 April 1969
Funding AgencyGrant Number
Woodrow Wilson FoundationUNSPECIFIED
Office of EducationUNSPECIFIED
Record Number:CaltechTHESIS:02012016-080841282
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9554
Deposited On:02 Feb 2016 16:39
Last Modified:26 Apr 2024 21:01

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