= Solution
For parallel propagation, use the undivided <dispersion relation>:
$$
P(n^2-R)(n^2-L)=0.
$$
The five requested cases are:
* \b[<Plasma oscillations>:] $P=0$ and $\delta\mathbf E\parallel\mathbf B_0\parallel\mathbf k$. With negligible <ion> response, $\omega^2=\omega_{pe}^2$, independently of $k$ in this cold approximation.
* \b[<Cold-plasma right-handed waves>:] $n^2=R$, $E_y=iE_x$, $E_z=0$, for propagation along $+\mathbf B_0$. At a fixed point, the real <electric field> rotates from $+\hat{\mathbf x}$ towards $+\hat{\mathbf y}$, in the sense of <electron> gyration.
* \b[<Cold-plasma left-handed waves>:] $n^2=L$, $E_y=-iE_x$, $E_z=0$. The real <electric field> rotates in the opposite sense.
* \b[<Ordinary plasma waves>:] for perpendicular propagation, $\theta=\pi/2$, $n^2=P$, and $\delta\mathbf E\parallel\mathbf B_0$. Their <dispersion relation> is $\omega^2=\omega_{pe}^2+c^2k^2$.
* \b[<Extraordinary plasma waves>:] for perpendicular propagation, $n^2=RL/S$, $E_z=0$, and $E_y=igE_x/(S-n^2)$. In general the <electric field> has both longitudinal and transverse components.
For the perpendicular cases the undivided <dispersion relation> factors as $(n^2-P)(Sn^2-RL)=0$, up to an overall nonzero factor. This also identifies the <extraordinary plasma wave> <plasma wave resonance> at $S=0$.
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