I am reading Section: Polynomials in Several Indeterminates from textbook Analysis I by Herbert Amann and Joachim Escher.
Previously, the authors define $X^\alpha_\beta$ as follows
In Remark (c), I could not understand why $p = \sum_\alpha p_\alpha X^\alpha$ can be written in the form $p = \sum_{j=0}^n q_j X^j_m$ for suitable $n \in \Bbb N$ and $q_j \in K[X_1,\cdots,X_{m-1}]$.
In my understanding, $X^j_m = \begin{cases}1, &j=m \\ 0, &j\neq m \end{cases}$. As a result, $X^j_m \in K$.
I could not understand how the product $q_j X^j_m$ makes sense.
Please shed me some light. Thank you so much!
Update: I have added screenshots of this entire section so that you can grasp authors' ideas more easily.
The $X_i$ are indeterminates and do not have specific values. In the proof, the approach is to factor out the highest power of $X_m$ in each term so that you have a polynomial in one variable, namely $X_m$, except that the coefficients are in $K[X_1,\ldots,X_{m-1}]$. We are identifying $K[X_1,\ldots,X_{m-1}][X_m] $ with $K[X_1,\ldots,X_{m} ]$.