  Context. The transition from left-hand to right-hand polarized Alfv\'en waves
depends on the wavenumber, the ratio of kinetic to magnetic pressure $\beta$,
temperature anisotropy, and ion composition of the plasma. Along with the
temperature anisotropy, the electron-to-proton temperature ratio $T_e/T_p$ is
of great relevance for the characterization of the thermal properties of a
plasma. This ratio varies significantly between different space plasma
environments. Thus, studying how variations on this ratio affect the
polarisation properties of electromagnetic waves becomes highly relevant for
our understanding of the dynamics of space plasmas. Aim. We present an
extensive study on the effect of the thermal properties of electrons on the
behaviour and characteristics of Alfv\'enic waves in fully kinetic linear
theory, as well as on the transition from electromagnetic ion-cyclotron (EMIC)
to kinetic Alfv\'en waves (KAW). Method. We solve the fully kinetic dispersion
relation for oblique electromagnetic waves of the Alfv\'en branch in a
homogenous Maxwellian electron-proton plasma. We quantify the effect of the
thermal properties of electrons by varying the electron-to-proton temperature
ratio for different configurations of the propagation angle, $\beta_p=8\pi
nkT_p/B^2$, and wavenumber. Results. We show that the temperature ratio
$T_e/T_p$ has strong and non-trivial effects on the polarisation of the
Alfv\'enic modes, especially at kinetic scales and $\beta_e+\beta_p>0.5$. We
conclude that electron inertia plays an important role in the kinetic scale
physics of the KAW in the warm plasma regime, and thus cannot be excluded in
hybrid models for computer simulations.
