In [4] we have introduced a new distance between Galois orbits over $\mathbb{Q}$. Using generalized divisors, we have extended the notion of trace of an algebraic number to other transcendental quantities. In this article we continue the work started in [4]. We define the critical function for a class of transcendental numbers, that generalizes the notion of minimal polynomial of an algebraic number. Our results extend the results obtained by Popescu et al [5].

Given a prime number 𝑝 and the Galois orbit $O(x)$ of a normal element 𝑥 of $\mathbb{C}_p$, the topological completion of the algebraic closure of the field of 𝑝-adic numbers, we study the Iwasawa algebra of $O(x)$ with scalars drawn from $\mathbb{Q}_p$ and relate it with $\mathbb{Q}_p$-distributions and functionals.