  Let $\phi:\,X\rightarrow Y$ be a (possibly ramified) cover between two
algebraic curves of positive genus. We develop tools that may identify the Prym
variety of $\phi$, up to isogeny, as the Jacobian of a quotient curve $C$ in
the Galois closure of the composition of $\phi$ with a well-chosen map
$Y\rightarrow \mathbb{P}^1$. This method allows us to recover all previously
obtained descriptions of a Prym variety in terms of a Jacobian that are known
to us, besides yielding new applications. We also find algebraic equations for
some of these new cases, including one where $X$ has genus $3$, $Y$ has genus
$1$ and $\phi$ is a degree $3$ map totally ramified over $2$ points.
