One application on hypergeometic series and values of $g$-adic functions algebraic independence investigation methods

The article takes a look at transcendence and algebraic independence problems, introduces statements and proofs of theorems for some kinds of elements from direct product of $p$-adic fields and polynomial estimation theorem. Let $\mathbb{Q}_p$ be the $p$-adic completion of $\mathbb{Q}$, $\Omega_{p}$ be the completion of the algebraic closure of $\mathbb{Q}_p$, $g=p_1p_2\ldots p_n$ be a composition of separate prime numbers, $\mathbb{Q}_g$ be the $g$-adic completion of $\mathbb{Q}$, in other words $\mathbb{Q}_{p_1}\oplus\ldots\oplus\mathbb{Q}_{p_n}$. The ring $\Omega_g\cong\Omega_{p_1}\oplus\ldots\oplus\Omega_{p_n}$, a subring $\mathbb{Q}_g$, transcendence and algebraic independence over $\mathbb{Q}_g$ are under consideration. Also, hypergeometric series
$$f(z)=\sum\limits_{j=0}^{\infty}\frac{(\gamma_1)_j\ldots(\gamma_r)_j}{(\beta_1)_j\ldots(\beta_s)_j}(zt)^{tj},$$
and their formal derivatives are under consideration. Sufficient conditions are obtained under which the values of the series $f(\alpha)$ and formal derivatives satisfy global relation of algebraic independence, if $\alpha=\sum\limits_{j=0}^{\infty}a_{j}g^{r_{j}}$, where $a_{j}\in \mathbb Z_g,$ and non-negative rationals $r_{j}$ increase strictly unbounded.