  We analyze the instability of a vortex column in a dilute polymer solution at
large $\textit{Re}$ and $\textit{De}$ with $\textit{El} =
\textit{De}/\textit{Re}$, the elasticity number, being finite. Here,
$\textit{Re} = \Omega_0 a^2/\nu_s$ and $\textit{De} = \Omega_0 \tau$ are,
respectively, the Reynolds and Deborah numbers based on the core angular
velocity ($\Omega_0$), the radius of the column ($a$), the solvent-based
kinematic viscosity ($\nu_s = \mu_s/\rho$), and the polymeric relaxation time
($\tau$). The stability of small-amplitude perturbations in this distinguished
limit is governed by the elastic Rayleigh equation whose spectrum is
parameterized by $ \textrm{E} = \textit{El}(1-\beta)$, $\beta$ being the ratio
of the solvent to the solution viscosity. The neglect of the relaxation terms,
in the said limit, implies that the polymer solution supports undamped elastic
shear waves propagating relative to the base-state flow. The existence of these
shear waves leads to multiple (three) continuous spectra associated with the
elastic Rayleigh equation in contrast to just one for the original Rayleigh
equation. Further, unlike the neutrally stable inviscid case, an instability of
the vortex column arises for finite E due to a pair of elastic shear waves
being driven into a resonant interaction under the differential convection by
the irrotational shearing flow outside the core. An asymptotic analysis for the
Rankine profile shows the absence of an elastic threshold; although, for small
E, the growth rate of the unstable discrete mode is transcendentally small,
being O$(\textrm{E}^2e^{-1/\textrm{E}^{\frac{1}{2}}})$. An accompanying
numerical investigation shows that the instability persists for smooth
vorticity profiles, provided the radial extent of the transition region (from
the rotational core to the irrotational exterior) is less than a certain
$\textrm{E}$-dependent threshold.
