I saw the following exercise in Sharp's text book:
Let $K$ be a field, and let $R = K[x_1,...,x_n]$ be the ring of polynomials over $K$ in indeterminates $x_1,...,x_n$; let $a_1,...,a_n \in K$. Show that, in $R$, $$0 \subset (x_1 - a_1) \subset (x_1 - a_1, x_2-a_2) \subset ... \subset (x_1-a_1,..., x_n - a_n)$$ is a (strictly) ascending chain of prime ideals.
My attempt: I will show that $ J_1 = (x_1 - a_1),...,J_n=(x_1-a_1,..., x_n - a_n)$ are a maximal ideals of $R$. I already know that $J_n$ is a maximal ideal of $R$, the problem is $J_i,\ \text{for}\ i \in \{1,...,n-1\}$.
Let $I$ be a ideal of $R$ such that $J_1 \subset I \subseteq R$, i.é, exist $f \in I$ such that $f \notin J_1$. We have that
$$f = q(x_1-a_1) + r \hspace{2cm}(1)$$
for some $q, r \in R$ with $\text{degree}(r) < \text{degree}(x_1-a_1) = 1$ or $r = 0$. Since $f \notin J$ we have $\text{degree}(r) = 0 \hspace{3cm} (2)$
So...??
My questions:
for (1): Since $K$ is a field we have that $R$ is a Euclidean Domain and then we can apply the algorithm in (1), right??
for (2): What can we say about $r$, since $\text{degree}(r) = 0$? I got a feeling that is not true that $r \in K$.
Maybe this approach is not correct, but I can not think of any other approach by using the other characterization of prime ideals : $P$ is a prime ideal of $R$ if, and only if, $R/P$ is a integral domain.
Quotienting out by any of these ideals, say $(x_1-a_1,...,x_k-a_k)$, gives the ring $K[x_{k+1},...,x_n]$, which is an integral domain. I'll leave the details to you.