Abstract: We apply the theory of cyclotomy to find proper divisible difference sets and difference sets or to prove their nonexistence for certain important situations in the additive group $G$ of a finite field (i.e. $G=GF(p^\alpha),+$ where $p$ is an odd prime) and in the additive group $G$ of the direct sum of two finite fields (i.e. $G=GF(p^\alpha)\oplus GF(q^\beta),+$ where $p$ and $q$ are distinct odd primes). We show that for any multiplicative cyclic subgroup $C_0$ in $GF(p^\alpha)$ , neither $C_0$ nor $C_0\cup \{0\}$ forms a proper divisible difference set in $G=GF(p^\alpha),+$ relative to any additive subgroup $N$. We generalize the concept of divisible difference sets to that of separable difference sets. In $GF(p^\alpha)\oplus GF(q^\beta)$, we take $D$ to be a maximum multiplicative cyclic subgroup $C_0$. We obtain a necessary and sufficient condition for $C_0$ to be a separable difference set relative to some subset $S\subset G$ and necessary and sufficient conditions for $C_0$ to be a divisible difference set relative to some suitable subgroup $N$ for general $e$, the index of $C_0$ in the unit group of $GF(p^\alpha)\oplus GF(q^\beta)$. We study the special cases $e = 2$ and $e = 4$. We establish some necessary and sufficient conditions and obtain several families of proper divisible difference sets, including the family of twin prime power divisible difference sets. This result of twin prime power divisible difference sets parallels the classical result of Stanton and Sprott's twin prime power Hadamard difference sets. We show that these families of divisible difference sets are the only separable difference sets $C_0$ can form when $e = 2$. We study $C_0\cup \{0\}$ in a similar way and obtain some more families of proper divisible difference sets and other similar results. We then consider $C_0\cup \cal A$ for a suitable $\cal A$ in $G=GF(p^\alpha)\oplus GF(q^\beta),+$. We obtain a necessary and sufficient condition for $C_0\cup \cal A$ to be a separable difference set relative to some $S\subset G$. We show that if $C_0\cup \cal A$ is a divisible difference set in $G$ relative to a suitable subgroup $N$ then $C_0\cup \cal A$ is a difference set in $G$. We establish a necessary and sufficient condition for $C_0\cup \cal A$ to be a difference set in $G$ for general $e$. We study the special case $e = 2$ and show that the complement of $C_0\cup \cal A$ in $G$ gives a family of difference sets in $G$ which has the same parameters as the classical Stanton and Sprott's twin prime power Hadamard difference set family. We also obtain a family of separable difference sets. We show that the above two families are the only separable difference sets $C_0\cup \cal A$ can form when $e = 2$. We also study the special case when $e = 4$ and show that $C_0\cup \cal A$ gives only one difference set in $G$ when $e = 4$, a $(45, 12, 3)$-difference set.