I want to solve this prob "Let $K$ be a field. Let $X_1,...,X_n$ be indeterminates. Prove that the ideal generated by $(X_1-t_1)^{s_1},...,(X_r-t_r)^{s_r}$ is a primary ideal for all $0\leq r \leq n$, for all $t_1,...,t_r \in K$ and all $s_1,...,s_r\in \mathbb{N}$ ".

My attempt: as the problem seems hard to approach in my opinion, I want to prove a prob which looks more simple but is definitely a stronger result

"Let $R$ be an integral domain. Let $I$ be a primary ideal of $R$. THen $I[X]$ is a primary ideal of $R[X]$", which is exactly the name of of this post.

If this result is true, then by induction and the fact that an ideal whose radical is maximal is primary, we solve the original problem.

I tried to prove the second prob by definition of primary ideal, but failed.

Can you help me solve the second prob? THere is a chance this result is wrong, so can you show me the right way to solve Prob 1? THank you

I want to solve this problem:

Let $K$ be a field. Let $X_1,\ldots,X_n$ be indeterminates. Prove that the ideal generated by $(X_1-t_1)^{s_1},\ldots,(X_r-t_r)^{s_r}$ is a primary ideal for all $0\leq r \leq n$, for all $t_1,\ldots,t_r \in K$ and all $s_1,\ldots,s_r\in \mathbb{N}$.

My attempt: as the problem seems hard to approach in my opinion, I want to prove a problems which looks more simple but is definitely a stronger result:

Let $R$ be an integral domain. Let $I$ be a primary ideal of $R$. Then $I[X]$ is a primary ideal of $R[X]$.

which is exactly the name of of this post.

If this result is true, then by induction and the fact that an ideal whose radical is maximal is primary, we solve the original problem.

I tried to prove the second problem by definition of primary ideal, but failed.

Can you help me solve the second problem? There is a chance this result is wrong, so can you show me the right way to solve Prob 1? Thank you.