On the P-class groups of the pure number field Q(N 1/P) and its Galois closure Q(N 1/P, [stigma]P)

We study the class groups of the fields $K$ = $\mathbb{Q}(N^{1/p})$ and $M$= $\mathbb{Q}(N^{1/p}, ?_p)$, where $N$ and $p$ are primes and $?_p$ is a primitive $p^{th}$ 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 $\mathbb{F}_p$. $\\$ 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 $M/\mathbb{Q}(?_p)$, 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, $N \neq p$, and let $f$ denote the minimal positive integer $x$ such that $N^x$ ? 1 (mod $p$). Let ${U_{\mathbb{Q}(?p)}}^+$ be the real units of $\mathbb{Q}(?_p)$. Suppose $\frac{p-1}{f}$ is odd. Then ${U_{Q(?_p)}}^+ \subseteq N_{M/\mathbb{Q}(?_p)}(M^\ast)$, where $N_{M/\mathbb{Q}(?_p)}$ denotes the usual norm map of the extension $M/\mathbb{Q}(?_p)$.