= Solution
Here $\alpha$ is the <thermal expansion coefficient> and $\rho$ is the <mass density>. Let $K=|\mathbf k|$, $h^2=K^2-k_z^2$, $q=\mathbf B\cdot\mathbf k$, and $a^2=q^2/(\mu_0\rho)$, the squared <Alfvén frequency>. We initially consider a nonzero frequency. The temperature and magnetic equations for a <Fourier mode> give
$$
\widehat\theta=\frac{i\beta}{\omega}\widehat w,\qquad
\widehat{\mathbf b}=-\frac{q}{\omega}\widehat{\mathbf u},\qquad
w=u_z.
$$
In particular, the minus sign in the magnetic relation follows from $-i\omega\widehat{\mathbf b}=iq\widehat{\mathbf u}$. Substitution into the <magnetohydrodynamic momentum equation>, with $P=p/\rho$, gives
$$
-i\Delta\widehat{\mathbf u}+2\Omega\widehat{\mathbf z}\times\widehat{\mathbf u}
=-i\mathbf k\widehat P+g\alpha\widehat\theta\widehat{\mathbf z},\qquad
\Delta=\omega-\frac{a^2}{\omega}.
$$
For <incompressible flow>, $\nabla\times(\widehat{\mathbf z}\times\mathbf u)=-\partial_z\mathbf u$. Taking the vertical component of the <curl> eliminates <pressure> and gives, with $\widehat\zeta=(i\mathbf k\times\widehat{\mathbf u})_z$,
$$
-i\Delta\widehat\zeta-2i\Omega k_z\widehat w=0,
\qquad \Delta\widehat\zeta+2\Omega k_z\widehat w=0.
$$
Taking another <curl> and its vertical component uses $\nabla\times\nabla\times\mathbf u=-\nabla^2\mathbf u$ and $(\nabla\times\nabla\times(\theta\widehat{\mathbf z}))_z=h^2\widehat\theta$. Thus
$$
-i\Delta K^2\widehat w-2i\Omega k_z\widehat\zeta=g\alpha h^2\widehat\theta,
$$
or, after eliminating temperature,
$$
\left(\Delta K^2+\frac{g\alpha\beta h^2}{\omega}\right)\widehat w+2\Omega k_z\widehat\zeta=0.
$$
The two equations for vertical velocity and vertical <vorticity> have a nontrivial solution only when their determinant vanishes. This proves the <stratified magneto-Coriolis wave> <dispersion relation>
$$
\boxed{\Delta^2K^2-4\Omega^2k_z^2+
\frac{g\alpha\beta}{\omega}\Delta h^2=0,\qquad
\Delta=\omega-\frac{(\mathbf B\cdot\mathbf k)^2}{\mu_0\rho\omega}}.
$$
No division by $\Delta$ was necessary, so a possible $\Delta=0$ is retained. Division by $\omega$, however, was necessary. Stationary disturbances must be checked directly in the original equations: for example, with $q\ne0$, $\mathbf u=0$ and an appropriate solenoidal magnetic perturbation can balance temperature-induced <buoyancy> and <pressure>. Such static balances are separate from the nonzero-frequency wave branches.
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