I have asked a question here and gave a comment regarding an answer by Captain Lama. He seems to be busy or not interested in my question any more that he has not responded to my comment, so I have no choice but to post another related question in this thread.
I think it is not good to copy relevant concept/information (which I typed carefully) from that question to this one. The copying will make this thread unnecessarily long. If you are interested in answering this question, please take a look at the old one.
My textbook states that
$p=\sum_{\alpha} p_{\alpha} X^{\alpha} \in K\left[X_{1}, \ldots, X_{m}\right]$ can be written in the form $p=\sum_{j=0}^{n} q_{j} X_{m}^{j}$ for suitable $n \in \mathbb{N}$ and $q_{j} \in K\left[X_{1}, \dots, X_{m-1}\right]$.
From Captain Lama's answer here, I got that $X^j_m \in K[X_1,\dots,X_m]$ and $q_j \in K[X_1,\cdots,X_{m-1}]$.
My question: $K[X_1,\dots,X_m] \neq K[X_1,\cdots,X_{m-1}]$ and thus they are two different rings of polynomials. How can the product $q_{j} X_{m}^{j}$ make sense?
Thank you for your help!
What is happening here is that the polynomial $p \in K[X_1, \ldots, X_m]$ is being written as a polynomial in the one variable $X_m$, whose coefficients are polynomials in $K[X_1, \ldots, X_{m-1}]$. So $q_j X_m^j$ is not a product of elements of different rings, but rather a coefficient $q_j$ times a variable $X_m$ raised to a power, and a polynomial ring can have coefficients in any ring. So the fact that we can make polynomials of the form $\sum_{j = 1}^{n} q_j X_m^j$ makes sense for the same reason that it makes sense to consider polynomials with coefficients in say, the integers, or any arbitrary ring.
In other words, what this lemma is saying is we can write $K[X_1, \ldots, X_m]$ as $K[X_1, \ldots, X_{m - 1}][X_{m}]$, or more succinctly as $R[X_m]$, where $R$ is the ring $K[X_1, \ldots, X_{m - 1}]$.