Let $K\subset L$ be a field extension and let $I$ be an ideal of the polynomial ring $K[X_1,...,X_n]$. I want to show that if $1\in J=IL[X_1,...,X_n]$ then $1\in I$. ($J$ is the ideal of $L[X_1,...,X_n]$ generated by the elements of $I$.)
For the case $K$ is an algebraically closed field, i suppose that $1\notin I$ then there exist a maximal ideal $m$ of $K[X_1,...,X_n]$ such that $I\subset m$ and we know that in this case $m=(X_1-a_1,...,X_n-a_n)$ for some $(a_1,...,a_n)\in K^n$ and we continue in this way we get a contradiction... Now i don't know what to do if $K$ is not necessarily algebraically closed field. Thanks for your help.
Hint:
$L[X_1,\dots,X_n]$ is a faithfully flat $K[X_1,\dots,X_n]$-algebra, hence $$IL[X_1,\dots,X_n]\cap K[X_1,\dots,X_n]=I.$$