I have a problem in understanding the proof of the following theorem:
Let $I\subseteq\mathbb{C}[[x_1,...,x_n]]$ be an ideal. Then there exists a $k\in\mathbb{N}$ and a linear coordinate change $\phi:\mathbb{C}[[x_1,...,x_n]]\to\mathbb{C}[[x_1,...,x_n]]$ such that $\mathbb{C}[[x_1,...,x_k]]\subseteq\mathbb{C}[[x_1,...,x_n]]/\phi(I)$ and $\mathbb{C}[[x_1,...,x_n]]/\phi(I)$ is finite as $\mathbb{C}[[x_1,...,x_k]]$-module.
As for the proof, assume $I\neq0$. Let $0\neq f\in I$, then one finds a linear coordinate change $\phi_1:\mathbb{C}[[x_1,...,x_n]]\to\mathbb{C}[[x_1,...,x_n]]$ such that $\phi_1(f)$ is $x_n$-regular. By the Weierstraß Preparation Theorem, there is a unit $u$ and a Weierstraß polynomial $p$ w.r.t. $x_n$ such that $u\phi_1(f)=p$. In particular, $\mathbb{C}[[x_1,...,x_{n-1}]]\hookrightarrow\mathbb{C}[[x_1,...,x_n]]/p$ is finite. Hence $\mathbb{C}[[x_1,...,x_{n-1}]]\to\mathbb{C}[[x_1,...,x_n]]/\phi_1(I)$ is finite.
The bold part is where I am still stuck. First, finite means the right hand side is a finitely generated module over the left hand side, is this correct (or is it over the image of the lhs)? Assuming this, the first part is due to $p$ being a Weierstraß polynomial w.r.t. $x_n$, i.e. $p$ has terms containing $x_n$ only up to a certain order.
Edit: If $f=\sum_{\mu\geq m}f_\mu$ is the homogeneous decomposition of $f$ with $f_m\neq 0$, take any $(a_1,...,a_{n-1})\in\mathbb{C}^{n-1}$ with $f_m(a_1,...,a_{n-1},1)\neq 0$, and define $\phi_1(x_i):=x_i+a_i x_n$ for i
And does it have anything to say that once the arrow $\hookrightarrow$ is used (which indicates injectivity to me), and in the next sentence it's just a 'normal' map?
Thank you in advance!