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Let $\mathbb{F}$ be a field, and consider $\mathbb{F}[[x]]$, the ring of formal power series with coefficients in $\mathbb{F}$, i.e. the set of expressions of the form $$\sum_{n=0}^{\infty}a_n x^n,\quad a_n\in\mathbb{F}$$ with the usual rules for addition and multiplication. How to show that any element that does not belong to the principal ideal $\langle x\rangle$ is invertible?

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1 Answer 1

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Let $g \not \in \langle x \rangle$. This means that the constant term is non-zero. Assume $g=\sum_{n=0}^\infty b_n x^n$. We want to find an inverse $f$ to $g$. Assume $f=\sum_{n=0}^\infty a_n x^n$. We want to solve

$$ \left(\sum_{n=0}^\infty b_nx^n\right) \left(\sum_{n=0}^\infty a_nx^n \right) = 1$$

Now, the constant term in the expansion is $b_0a_0$, which must equal $1$, so $a_0=1/b_0$. The coefficient of $x$ is $b_0a_1+b_1a_0=0$. So $a_1=-b_1a_0/b_0$. We can continue this way (prove this) to find all the $a_i$.

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