Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question

1 Answer 1

up vote 1 down vote accepted

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 $a_i$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.