We study the class groups of the fields $K$ = $Q(N1/p)$ and $M$= $Q(N1/p,\zeta p)$, where $N$ and $p$ are primes and $\zeta p$ 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 $SF$ . Define $rank(SF)$ as the dimension of $SF/(SF)p$ as a vector space over $Fp$. $$ 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 $rank(SM)$ = 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 $M/Q(\zeta p)$, and use these results to give bounds on $rank(SM)$. For example, we show the following: Proposition 5.2. Let $p$ and $N$ be any primes with $p$ ≥ 5, $N\ne p$, and let $f$ denote the minimal positive integer $x$ such that $Nx$ ≡ 1 (mod $p$). Let $UQ(\zeta p)+$ be the real units of $Q(\zeta p)$. Suppose $p-1f$ is odd. Then $UQ(\zeta p)+\subseteq NM/Q(\zeta p)(M\ast )$, where $NM/Q(\zeta p)$ denotes the usual norm map of the extension $M/Q(\zeta p)$.