Radu Murgescu
Radu Murgescu

Ph.D. (2009) Cornell University

First Position
Dissertation
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Abstract: We study the class groups of the fields K = Q(N^{1/p}) and M = Q(N^{1/p}, ζ_{p}), where N and p are primes and ζ_{p} is a primitive pth root of unity. Furthermore, we restrict ourselves to the study of the pclass groups, i.e. the Sylow psubgroups of the class groups in question. We denote the pclass group of a field F by S_{F}. Define rank(S_{F}) as the dimension of S_{F} / (S_{F})^{p} as a vector space over F_{p}.
Frank Gerth III, in [9], settled a problem left open by F. Calegari and M. Emerton in [4]. He, in turn, posed a related question, which we answer in this thesis, as follows.
Theorem 3.15. Let N ≡ 4 or 7 (mod 9), and suppose 3 is a cubic residue (mod N). Then rank(S_{M}) = 2.
For general values of p, we obtain results (Propositions 2.13, 4.1, 4.3, 5.2, 5.3, and 5.9) on the existence of certain norms in the extension M/Q(ζ_{p}), and use these results to give bounds on rank(S_{M}). For example, we show the following:
Proposition 5.2. Let p and N be any primes with p ≥ 5, N ≠ p, and let f denote the minimal positive integer x such that N^{x} ≡ 1 (mod p). Let U_Q(ζ_{p})^{+} be the real units of Q(ζ_{p}). Suppose (p – 1)/f is odd. Then U_Q(ζ_{p})^{+} ⊆ N_{M/Q(ζ_{p})}(M*), where N_{M/Q(ζ_{p})} denotes the usual norm map of the extension M/Q(ζ_{p}).