Likelihood function ARCH models

Question

Suppose that the values of {$\epsilon_t$} are drawn from a normal distribution having a mean of zero and a constant variance $\sigma^2$, then the likelihood function is: $$L_t=\frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-\epsilon^2_t}{2\sigma^2}}$$ My textbook now sets $\epsilon_t$ as an ARCH(1) error with $\epsilon_t=\nu_t\sqrt{h_t}$ where $\nu_t$ is a white noise and it defines its likelihood as before $$L_t=\frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-\epsilon^2_t}{2\sigma^2}}$$ Shouldn't this be the likelihood function only in the case of Normal distribution? In the case of ARCH model do I have hypothesis of Gaussian distribution for $\epsilon_t$?