How can I show that the ideal $(x_1,x_2,x_3,...)$ (with infinitely many variables) in the ring $K[x_1,x_2,x_3,...]$ is not finitely generated.
I cannot complete the arguments as I thought if the ideal is generated by finitely many polynomials say $f_1,f_2,........f_n$ then these polys will contain only finitely many variables. Let $x_k$ be the variable with highest subscript which will be involved in $f_i 's$. So $x_{k+1}$ will not appear in any of the $f_i's$. My target is to show that $x_{k+1}$ cannot be written as $K[x_1,x_2,x_3,.....]$ linear combination of $f_i's$. If possible suppose $x_{k+1}$=$g_1.f_1 +g_2.f_2+.........+g_n.f_n$. I am trying by putting $x_1=x_2=...=x_k$=$1$ but then I can't argue further. I want this to be completed in this way. Help me. Thank You.
