For the past couple of days i had been working on an interesting homework problem, my interpretation of which is as follows:
Let $m$ be a maximal ideal of $k[x_1,\cdots,x_n]$. Let $m_i = m \cap k[x_i], i=1,\cdots,n$. Then show that $m$ is the unique pre-image of $m_1,\cdots,m_n$, under the map $\operatorname{Specm} k[x_1,\cdots,x_n] \rightarrow \prod_i \operatorname{Specm} k[x_i]$ given by $m \mapsto (m_i)_i$, if and only if $\left[k[x_1,\cdots,x_n]/m:k\right] = \prod_i \left[k[x_i]/m_i :k\right]$.
Related to this problem are the two questions of mine: Intersecting maximal ideals of $k[x_1,\dots,x_n]$ with $k[x_i]$ and Conditions for the degree of field extension reaches its upper bound.
PS: My homework is submitted, please feel free to attack to the problem :)