In the lecture, we proved the implicit function theorem:
Let $f\in\mathbb{C}[[x_1,...,x_n,y]]$ such that $f(0)=0$ and $\frac{\partial f}{\partial y}(0)\neq 0$. Then there exists a unique $\phi\in\langle x_1,...,x_n\rangle\mathbb{C}[[x_1,...,x_n]]$ with $$f(x_1,...,x_n,y)=0\Longleftrightarrow y=\phi(x_1,...,x_n).$$
One can use this to show e.g. $-1+\sqrt{x+1}\in\mathbb{C}[[x]]$ by considering $f=(y+1)^2-x-1$ in $\mathbb{C}[[x,y]]$. In particular, we used it in the proof of the Morse lemma:
Let $f\in\mathfrak{m}^2\subseteq\mathbb{C}[[x_1,...,x_n]]$ such that $(\frac{\partial^2 f}{\partial x_i\partial x_j}(0))$ has maximal rank. Then $f$ is right equivalent to $x_1^2+...+x_n^2$, i.e., there exists an automorphism $\phi$ of $\mathbb{C}[[x_1,...,x_n]]$ s.t. $\phi(f)=x_1^2+...+x_n^2$.
As for the proof, we write $$f=\sum_{i,j}x_ix_jH_{ij},$$ and we may suppose $H_{ij}=H_{ji}$ for all $i,j$. The rank of $(\frac{\partial^2 f}{\partial x_i\partial x_j}(0))$ is equal to the rank of $(H_{ij}(0))$. By induction on $s$, we suppose we found coordinates $(y_1,...,y_n)$ such that $$f(y_1,...,y_n)=y_1^2+...+y_{s-1}^2+\sum_{i,j\geq s}x_ix_jH_{ij}(y_1,...,y_n).$$ By assumption, at least on of the $H_{ij}(0)$ is non-zero, and we may assume $H_{ss}(0)\neq 0$ after a linear change of coordinates in $y_s,...,y_n$. By the implicit function theorem, consider the square root $g=\sqrt{H_{ss}}$ which is in fact a unit.
This last sentence is where I am stuck. Why can we consider this square root? What I tried to do was considering $t^2-H_{ss}\in\mathbb{C}[[y_1,...,y_n,t]]$, but then the assumptions of the implicit function theorem are not fulfilled, since $t^2-H_{ss}$ is not $t$-regular of order $1$.
I also tried using something like $(t+1)^2-H_{ss}$, which would then show the existence of $\phi=\sqrt{H_{ss}}-1$ in $\mathbb{C}[[y_1,...,y_n]]$, hence also $\sqrt{H_{ss}}$ exists. But here, the other assumption of the IFT probably fails, namely $1-H_{ss}(0)$ could be $0$, since we only know $H_{ss}(0)\neq 0$. Is the solution to choose a unit $m$ with $m^2\neq H_{ss}(0)$ in $\mathbb{C}$, and consider $(t+m)^2-H_{ss}$? This should fulfill the assumptions, and yield $\sqrt{H_{ss}}-m\in\mathbb{C}[[y_1,...,y_n]]$. But it feels so 'indirect' and 'constructed' to me that I think it has to be wrong and that I'm just not seeing it though it should be simple... I don't seem to get how this should follow directly from the IFT, maybe you can help me out here.
Thank you in advance!
