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This is not the definition of the refractive index. Its definition is $n^2 = \epsilon$. The dielectric "constant" is in turn defined as $\epsilon = \epsilon_0 ( 1+ \chi_e) $ where the polarisation $P = \epsilon_0 \chi_e E$ is related to the applied field by the electric susceptibility. See https://en.wikipedia.org/wiki/Electric_susceptibility.