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I am currently studying the ring of formal power series over an arbitrary characteristic field K, and I'm trying to prove the following property:

Let $f = (f_{1},\cdots,f_{m}) \in \mathbb{K}[[x_{1},\cdots,x_{n}]]^{m}$ be a sequence of formal power series with no constant term. Then,

If $g \in \mathbb{K}[[y_{1},\cdots,y_{m}]]$, $g \mapsto g(f)$ (the substitution homomorphism) is a ring isomorphism $\iff$ The determinant of $\partial{f_{i}}/\partial{x_{j}(0,0)}$ is non zero.

The book I'm using to study this kind of properties says it is a consequence of the Inverse Function Theorem in the formal power series ring. However, I'm not able to prove neither of the implications.

Could you give me any help? Thanks in advance!

HenryPA
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  • It is not appropriate to repost your closed questions. If you believe your closed question should be reopened, the appropriate thing to do is to edit it. (I don't think your original question should be reopened at present, because the duplicate tells you exactly how to solve your problem. If you don't quite understand or need clarification, that would be a good thing to edit in!) – KReiser Nov 17 '21 at 19:01
  • The duplicate does not give me any clarification about my question. – HenryPA Nov 17 '21 at 19:12

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