We know that $\mathbb{C}$ (set of all complex numbers) is algebraically closed, so we have to think about non-algebraic extension. Now, I found that $\mathbb{C}(X)$, the field of rational fractions coefficients from $\mathbb{C}$ is the field that contains $\mathbb{C}$ as a proper subfield. $\mathbb{C}(X)$ is defined as : $$\mathbb{C}(X)=\left\{ \dfrac{p(X)}{q(X)} \ | \ (p(X), q(X)) \in \mathbb{C}[X]^2 \ \& \ q(X) \neq 0_{\mathbb{C}[X]} \right\}$$
My doubt is : Is $q(x)$ not identically zero or $q(x)$ has no zero in $\mathbb{C}$. Because we know every function (polynomial) has all it's zeros inside the field $\mathbb{C}$, by fundamental theorem of algebra.