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Let $\mathbb{K}$ be a field, and consider $\mathbb{K}[[x]]$, the ring of formal power series with coefficients in $\mathbb{K}$, i.e. the set of expressions of the form $$\sum_{n=0}^{\infty}a_n x^n,\quad a_n\in\mathbb{K}$$ with the usual rules for addition and multiplication. How to show that $\mathbb{K}[[x]]$ is local ring ?

Thank you in advance

Z. Alfata
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Hint: take an element with non-zero constant term, and construct an explicit inverse, degree by degree (or at least show that it can be done, by finding the first three or so terms of the inverse and point out that you can keep going indefinitely). This shows that $(x)$ is the only maximal ideal.

Arthur
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    Or: in case the constant term is $1$, so that the series is $1-xg(x)$, then the reciprocal would be $\sum_0^\infty(xg(x))^n$. The general case follows. – Lubin May 17 '17 at 18:39