= Solution
An <electrostatic plasma wave> has $\delta\mathbf E=-i\mathbf k\phi$. In the absence of externally imposed <charge density>, the <Gauss law> for the <magnetized cold-plasma dielectric tensor> is $\mathbf k\cdot\boldsymbol\epsilon\cdot\delta\mathbf E=0$. Consequently,
$$
\boxed{k_\perp^2S+k_\parallel^2P=0.}
$$
The antisymmetric terms of the <magnetized cold-plasma dielectric tensor> cancel in this scalar contraction. For oblique propagation this is the leading longitudinal condition in the large-<refractive index> approximation; a small transverse <electric field> is generally required by the full <Maxwell equations> at finite $n$. For parallel propagation an exactly longitudinal solution exists: $P=0$, hence $\omega^2\simeq\omega_{pe}^2$. It coincides with the $\theta\to0$ limit of the finite-coefficient <plasma wave resonance> condition in <1/c>. At exactly $\theta=0$, $P$ factors out of the transverse <dispersion relation>, leaving the independent <plasma oscillation> rather than an isolated divergence of the transverse branches.
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