Let $\mathbb{Q}$ be the field of rationals and $\alpha$ be an irrational number which is not algebraic over $\mathbb{Q}$. Then for arbitrary element $w \in \mathbb{Q}[\alpha]$, there exists an unique polynomial $P$ with $P(\alpha)=w$.
My question is the following:
Given $\alpha$ and $w$, is it possible to determine the polynomial $P$ with 'basic' operations in $\mathbb{R}$, such as addition, multiplication, square root, the limit of some sequences, or etc?
Thinking $\mathbb{Q}[\alpha]$'s basis as $\{1, \alpha, \alpha^2, \cdots \}$, then the problem is equivalent to:
Can we represent the ordinary inner product in $\mathbb{Q}[\alpha]$ with respect to basis $\{1, \alpha, \alpha^2, \cdots \}$ with 'basic' operations in $\mathbb{R}$ ?
If then, we can determine coefficients of $P$ by calculating the inner product between $w$ and $\alpha^n$.
If it cannot be represented by 'basic' operations, what other methods or algorithms can be tried to find $P$ or its coefficients?