How do I write down "the set of all rational functions in one variable over $ \mathbb C$"?

Question

What is the best way to write down "the set of all rational functions over $\mathbb{C}$"?

This is what I have tried:

$$ \mathbb{C}(t) = \left\{ \frac{P(t)}{Q(t)} \Big| t \in \mathbb{C}, \; P(t) = \sum_{i=0}^{n \in \mathbb{N}}a_{i}t^n, \; Q(t) = \sum_{j=0}^{k \in \mathbb{N}}b_{j}t^n\Big| \right\} $$

Answer 1

In the notation of abstract algebra: $$\Bbb C(z)$$