A topological group $G$ is said to be solenoidal if it contains a dense one-parameter subgroup. That is, there exists a continuous homomorphism from the real numbers into $G$ such that the image is dense. We can obtain information about a topological group by studying its character group, the set of all continuous homomorphisms from $G$ into the circle group $T$. This is analogous to studying the dual of a vector space in functional analysis. We show that a compact group is solenoidal if and only if its character group is topologically isomorphic with a subgroup of the real line with the discrete topology. Along the way, we encounter the$a$-adic numbers, the $a$-adic solenoid, and ultimately a corollary which tells us that all compact solenoids can be expressed in terms of an uncountable product of $a$-adic solenoids.