Arithmetic properties of direct product of $p$-adic fields elements, II

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. Here are appropriate theorems for numbers not only like $\alpha=\sum\limits_{j=0}^{\infty}a_{j}g^{r_{j}}$ where $a_{j}\in \mathbb Z_g,$ and non-negative rationals $r_{j}$ increase strictly unbounded. But, for numbers $f(\alpha)$, where $f(z)=\sum\limits_{j=0}^{\infty}c_jz^j\in\mathbb Z_g[[z]]$. Furthermore, let $\widehat{\mathbb Q}\cong\prod\limits_{p}\mathbb{Q}_p$ be the ring of polyadic numbers, then, the article takes a look at $\widehat{\Omega}=\prod\limits_{p}\Omega_p$, there are similar results for numbers like $f(\alpha)$, where $f(z)=\sum\limits_{j=0}^{\infty}c_jz^j\in\widehat{\mathbb Z}[[z]]$, $\alpha=\sum\limits_{k=1}^{\infty}a_{k}g^{r_{k}}$, $a_{k}\in \mathbb Z_g,$ $g=(p_1,\ldots,p_n,\ldots)$.