Let $S=\mathbb{C}[x_1,x_2,x_3,x_4]$ be a polynomial ring. Let $p_i=x_1^i+\cdots+x_4^i$ be the power sum symmetric polynomial in $\mathbb{C}[x_1,x_2,x_3,x_4]$. Let $I=(p_1,p_2)$ be an Ideal of $\mathbb{C}[x_1,x_2,x_3,x_4]$.

Question: If $(p_1,p_2)$ is a prime ideal, can we also say $(p_m,p_{2m})$ is also a prime ideal in $S$.

Known: We know that $p_1,p_2$ is a regular sequence in $S$. ( Since $p_1,p_2,p_3,p_4$ is a regular sequence in $\mathbb{C}[x_1,x_2,x_3,x_4]$, see Conca, Krattenthaller and Watanabe http://arxiv.org/abs/0801.2662. And subset of a regular sequence is also a regular sequence). By lemma 2.2 in Conca, Krattenthaller and Watanabe, one concludes $p_m,p_{2m}$ is a regular sequence in $\mathbb{C}[x_1,x_2,x_3,x_4]$.

Knowing $(p_1,p_2)$ is a prime ideal, Can one pass to say, $(p_m,p_{2m})$ is also a prime ideal in $S$. To prove $(p_1,p_2)$ is a prime ideal, one can use Serre Criterion, see Theorem 18.15 in Eisenbud, Algebra with a view towards algebraic geometry.

Thanks

Neeraj