Radu Murgescu

TBA

On the p-Class Groups of the Pure Number Field Q(N^1/p) and its Galois Closure Q(N^1/p, ζ_p)

Ravi Ramakrishna

Algebraic number theory

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 p-class groups, i.e. the Sylow p-subgroups of the class groups in question. We denote the p-class 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).

PhD

PhD