Arithmetic properties of direct product of p-adic fields elements

Abstract

The article considers the transcendence and algebraic independence problems,
introduce 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 $\Omega_g\cong\Omega_{p_1}\oplus\ldots\oplus\Omega_{p_n}$,
contains a subring $\mathbb{Q}_g$. The transcendence and algebraic independence over $\mathbb{Q}_g$
are under consideration. Here are appropriate theorems for numbers like
$\alpha=\sum\limits_{j=0}^{\infty}a_{j}g^{r_{j}},\;
\text{where}\;a_{j}\in \mathbb Z_g,$ and non-negative rational numbers $r_{j}$ increase
to strictly unbounded.