# Atiyah Macdonald Exercise 1.5

I'm trying to prove that:

The contraction of a maximal ideal $$mathfrak {m}$$ of $$A[[x]]]$$ is a maximal ideal of $$A$$, and $$mathfrak {m}$$ is generated by $$mathfrak {m} ^ c$$ and $$x$$.

I have made the following progress:

assume $$x not in mathfrak {m}$$. then $$1-xg in mathfrak {m}$$ for some people $$g in A[[x]]$$, and this is a contradiction because $$1-xg$$ is a unit. So, $$x in mathfrak {m}$$. Consider any non-zero element $$a in A$$. We want to show that there is $$b in A$$ such as $$ab equiv 1 ( mathfrak {m} ^ c)$$. Since $$A[[x]]/ mathfrak {m}$$ is a field we know that there is $$f = sum a_nx ^ n in A[[x]]$$ such as $$af equiv 1 ( mathfrak {m})$$, and since $$x in mathfrak {m}$$, $$aa_0 equiv 1 ( mathfrak {m})$$. How can I conclude from this that $$aa_0 equiv 1 ( mathfrak {m} ^ c)$$ ?

Thank you.