Non-solvable polynomial over a PID

Question

I am trying to find a good candidate for a non-solvable quadratic polynomial $f(x)=a(z)x^2+b(z)x+c(z)$ over the PID, $\mathbb{C}[[z]]$, of formal power series $a(z),b(z),c(z)$ with complex coefficients. I somehow thought that I can try to make $f$ irreducible, but I seem to get stuck since it seems solvability of polynomials only make sense over a field. Is it possible to somehow relate this polynomial with anything familiar, since the coefficients $a(z),b(z),c(z)$ of $f$ are rather unwieldy.