Find the height of prime ideal $p=(x_n-x_1^n,\ldots ,x_2-x_1^n)$ in $\mathbb{C}[x_1,\ldots,x_n]$

Question

Find $\operatorname{ht}(p)$ where $p=(x_n-x_1^n,\dots,x_2-x_1^n)$ ideal of $\mathbb{C}[x_1,\ldots,x_n]$.

$\operatorname{ht}(p)=$ height of a prime $p$

How to prove $p$ is prime ?

Answer 1

Use repeatedly $R[X]/(X-a)\simeq R$ ($a\in R$) and show that $\mathbb{C}[x_1,\ldots,x_n]/p\simeq\mathbb C[x_1]$, so $p$ is prime of height $n-1$.