I am asking about part (a) of Exercise 23, Chapter 5 in Serge Lang's Algebra. Let $k$ be a finite field of $q$ elements. The zeta function is defined to be $$Z(t)=(1-t)^{-1}\prod_p(1-t^{\deg p})^{-1}$$ where $p$ ranges over all monic irreducible polynomials over $k$. Prove that $Z(t)$ is a rational function.
My attempts so far: Let $\psi(d)$ be the number of monic irreducible polynomials of degree $d$. Then for each $d$, there are $\psi(d)$ irreducible polynomials of degree $d$, i.e. $(1-t^d)^{-1}$ appears $\psi(d)$ times in the product. Hence $$\prod_p(1-t^{\deg p})^{-1}=\prod_{d=1}^{\infty}(1-t^d)^{-\psi(d)}$$ But I can't seem to expand this product any further. Any help is appreciated. A hint is okay.