Ring of integers of $\mathbb{C}_p$

Question

Sorry, maybe this is a really stupid question. Let $\mathbb{C}_p$ be the completion of the algebraic closure $\overline{\mathbb{Q}_p}$ of the field of $p$-adic numbers. We know that there exists a way to extend the $p$-adic valuation to $\mathbb{C}_p$, and in this way it is possible to define the ring of integers $\mathcal{O}_{\mathbb{C}_p}$ given by elements of valuation greater or equal than $0$. It is also possible to define the maximal ideal of this valuation ring given by elements of valuation strictly greater than $0$. Now, is this ideal finitely generated?

Answer 1

No it isn't - indeed, suppose it were, say $\mathfrak{m}_{\mathbb{C}_p} = (x_1,\ldots,x_n)$. Then $v(x) \geq \min v(x_i) > 0$ for any $x\in\mathfrak{m}_{\mathbb{C}_p}$. However, this is a contradiction as there exist elements of $\mathfrak{m}_{\mathbb{C}_p}$ with valuation strictly less than $\min v(x_i)$ (for example, one of the $\sqrt{x_i}$).