Consider a field extension $L\subseteq K$ and suppose that $I=(f_1,\ldots,f_m)$ is an ideal of $L[T_1,\ldots,T_n]$. Denote with $I^e\subseteq K[T_1,\ldots,T_n] $ the extended ideal of $I$ through the canonical immersion $\iota:L[T_1,\ldots,T_n]\longrightarrow K[T_1,\ldots,T_n] $. Now I have to prove the following isomorphism of $K$-algebras
$$ \frac{L[T_1,\ldots,T_n]}{I}\otimes_L K\cong \frac{K[T_1,\ldots,T_n]}{I^e} $$
Maybe an explicit isomorphism can be given by $I+g\otimes a\longmapsto I^e +ag $, but I have problems to find the inverse. Any suggestion?
Thanks in advance
Firstly, note that $L[T_1,\ldots,T_n]$ is a $L$-vector space. You can see pretty easily that $L[T_1,\ldots,T_n]\otimes_L K\cong K[T_1,\ldots,T_n]$. Similarly, it should be clear that $I^e = I\otimes_L K$.
Consider the exact sequence of $L$-modules (ie, $L$-vector spaces) $$0\rightarrow I\rightarrow L[T_1,\ldots,T_n]\rightarrow L[T_1,\ldots,T_n]/I\rightarrow 0$$ Since $K$ is a flat $L$-module (any module over a field is flat), you get an exact sequence $$0\rightarrow I\otimes_L K\rightarrow L[T_1,\ldots,T_n]\otimes_L K\rightarrow (L[T_1,\ldots,T_n]/I)\otimes_L K\rightarrow 0$$ ie, you have an exact sequence $$0\rightarrow I^e\rightarrow K[T_1,\ldots,T_n]\rightarrow(L[T_1,\ldots,T_n]/I)\otimes_L K \rightarrow 0$$ which presents $(L[T_1,\ldots,T_n]/I)\otimes_L K$ as a quotient of $K[T_1,\ldots,T_n]$ by $I^e$.