Is a prime ideal in the polynomial ring over an algebraically closed field prime also in polynomial rings over extension fields?

Question

Let $k \subsetneq K$ a field extension, $k$ algebraically closed, $I \subset k[x_1,\ldots, x_n]$ a prime ideal generated by the polynomials $f_1, \ldots, f_r$. Let $J \subset K[x_1, \ldots, x_n]$ be the ideal generated by $f_1, \ldots, f_r$. Is $J$ then also prime?

Can you provide a proof or a counterexample (or a reference to a book where the proof could be found)?

Answer 1

$K[X_1,\dotsc,x_n]/J = K \otimes_k k[x_1,\dotsc,x_n]/I$ is the tensor product of two integral domains over an algebraically closed field, hence also an integral domain (see e.g. here). So we may replace $K$ by any $k$-algebra which is an integral domain.