Convergent power series and Euclidean domain, or field.

Question

Consider the following power series (as a function of $z$):

$$G(z) = \sum_{i=0}^\infty a_i z^{-i},$$

where $a_i$ is a real coefficient and $z$ is any complex number. It also satisfies that

$$\sum_{i=0}^\infty |a_i| < \infty.$$

Then consider the set $\mathcal{V}$ that contains all such $G(z)$ (as a function of $z$), and $\mathcal{V}$ can be equipped with the standard addition and the Cauchy product. Then it is not hard to show that $\mathcal{V}$ is a commutative ring.

The questions are about some further steps:

1. Does $\mathcal{V}$ form a Euclidean domain under some Euclidean function?

   I know that the set of formal power series forms a Euclidean domain under certain Euclidean function, but how about this case?

2. Does $\mathcal{V}$ form a field?

   I cannot show the inverse property for $\mathcal{V}$ under the Cauchy product. For any $G_1 \in \mathcal{V}$, is there another $G_2 \in \mathcal{V}$ such that the Cauchy product of $G_1$ with $G_2$ equals one (multiplicative identity)?

Thank you in advance for your answers.