I want to show that each of the ideals $I_k=\langle x_1,...,x_k\rangle$ in $F[x_1,...,x_n]$ for $1\leq k \leq n$ are prime, and that each of their powers is primary, where $F$ is a field.
I have been able to show that each $I_k$ is prime, by considering the map $F[x_1,...,x_n] \rightarrow F[x_{k+1},...,x_n]$ given by $f(x_1,...,x_n)\mapsto f(0,...,0,x_{k+1},...,x_n)$ and using the first isomorphism theorem to show that $F[x_1,...,x_n]/I_k \cong F[x_{k+1},...,x_n]$ is a domain.
The part where I'm struggling is showing that each power of the $I_k$ is primary. I managed to prove that when $n=3$, $I_2^2$ is primary using brute force, but I'm unable to generalise this approach to either $n>3$ or higher powers of the ideal. Any help would be appreciated!
Let there exist $f,g \in f[X_1,...,X_n]$ for which $fg\in I_k^m$ for some $m\geq 1$ and $f\notin I_k^m$ and $g\notin \sqrt{I_k^m}=I_k$. With this assumption we intend to arrive at a contradiction.
First note that $I_k^m=\langle X_1^{n_1}...X_k^{n_k}\mid n_1+...+n_k=m\rangle$.
By the assumption on $f$ and $g$ :
We can write $f=f_1+f_2+f_3$ and $g=g_1+g_2$, such that (i) $f_1\in I_k^m$, (ii) $0\neq f_2+f_3$, (iii) $f_2=$ an $F$-linear combination of monomials $X_1^{l_1}...X_n^{l_n}$ such that $l_1+...+l_k< m$, (iv) $f_3=$ an $F$-linear combination of monomials $X_{k+1}^{r_1}...X_n^{r_{n-k}}$
and,
(i) $g_1\in I_k$ and (ii) $0\neq g_2$ is an $F$-linear combination of monomials $X_{k+1}^{l_1}...X_n^{l_{n-k}}$.
Observe that $fg=f_1g+(f_2+f_3)g\in I_k^m\Rightarrow f_2g+f_3g_1+f_3g_2\in I_k^m\Rightarrow f_3g_2=0\Rightarrow f_3=0\Rightarrow f_2\neq 0$ and hence, $f_2g_1+f_2g_2\in I_k^m$
Now let $d=$min {$l_1+...+l_k\mid$ $X_1^{l_1}...X_n^{l_n}$ is a monomial in $f_2g_2$}. Since $f_2\neq 0$, we have $m>d>0$. Thus we note that none of the monomials $X_1^{l_1}...x_n^{l_n}$ in $f_2g_2$ with $l_1+...+l_k=d$ gets cancelled by the monomials occurring in $f_2g_1$, contradicting $f_2g_1+f_2g_2\in I_k^n$
This completes the proof.
However, if one needs a smarter argument which avoids the above computation one may follow the hint provided above by user 26857 or equivalently, refer to
Any power of $(X_1, X_2,...,X_i)$ is primary in $k[X_1, X_2,...,X_n].$