A ‘strong’ form of the Fundamental Theorem of Algebra Let $ n \in \mathbb{N} $ and $ a_{0},\ldots,a_{n-1} \in \mathbb{C} $ be constants. By the Fundamental Theorem of Algebra, the polynomial
$$
p(z) := z^{n} + \sum_{k=0}^{n-1} a_{k} z^{k} \in \mathbb{C}[z]
$$
has $ n $ roots, including multiplicity. If we vary the values of $ a_{0},\ldots,a_{n-1} $, the roots will obviously change, so it seems natural to ask the following question.

Do the $ n $ roots of $ p(z) $ depend on the coefficients in an analytic sort of way? More precisely, can we find holomorphic functions $ r_{1},\ldots,r_{n}: \mathbb{C}^{n} \to \mathbb{C} $ such that
  $$
z^{n} + \sum_{k=0}^{n-1} a_{k} z^{k} = \prod_{j=1}^{n} [z - {r_{j}}(a_{0},\ldots,a_{n-1})]?
$$

The definition of a holomorphic function of several complex variables is given as follows:

Definition Let $ n \in \mathbb{N} $ and $ \Omega \subseteq \mathbb{C}^{n} $ be a domain (i.e., a connected open subset). A function $ f: \Omega \to \mathbb{C} $ is said to be holomorphic if and only if it is holomorphic in the usual sense in each of its $ n $ variables.

The existence of $ r_{1},\ldots,r_{n}: \mathbb{C}^{n} \to \mathbb{C} $ that are continuous seems to be a well-known result (due to Ostrowski, perhaps?), but I am unable to find anything in the literature that is concerned with the holomorphicity of these functions.
Any help would be greatly appreciated. Thank you very much! 

To give a little expansion to @Andreas’s answer, let’s examine a little more closely the way the coefficients depend on the roots. Let’s take an $n$-tuple of roots, say $\rho=(\rho_1,\cdots,\rho_n)$ and form the corresponding $n$-tuple whose entries are the coefficients $a=(a_0,\cdots,a_{n-1})$ of the monic polynomial whose roots are the $\rho_i$’s. You have the map $C\colon\rho\mapsto a$, and you can ask what the Jacobian determinant is of this map, call it $J$. Then the fact is that $J^2$ is the discriminant of the polynomial $F(x)=x^n+\sum_0^{n-1}a_{n-i}x^i$, which as you probably know is a polynomial in the $a_i$’s. This fact makes it very clear, via the Inverse Function Theorem, how and when and why the roots depend on the coefficients.
