CounterexampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-08-13

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$x$ has no multiplicative inverse in $R ⟦ x ⟧$ although it is invertible in $K ((x))$ when $R = K$ is a field

Counterexample

Let $K$ be a field. The series $x$ is not a unit in $K ⟦ x ⟧$ , but its image is a unit in $K ((x))$ with inverse $x^{- 1}$ .

Facts & Assumptions

Given: The hypotheses and notation of the statement above.

[F1]

A formal power series is a unit exactly when its constant coefficient is a unit (A formal power series is a unit exactly when its constant coefficient is a unit).

[F2]

Power series embed in Laurent series, and a nonzero Laurent series $h = x^{v_{x} (h)} u$ has inverse $x^{- v_{x} (h)} u^{- 1}$ ( $K ⟦ x ⟧$ embeds in $K ((x))$ as the nonnegative-order subring; every nonzero Laurent series is uniquely $x^{v_{x} (h)} u$ with $u \in K ⟦ x ⟧^{\times}$ and inverse $x^{- v_{x} (h)} u^{- 1}$ ).

Verification

technique · compare constant coefficient and Laurent order

1.1

The constant coefficient of $x$ is $0$ , which is not a unit in the field $K$ , so the unit criterion excludes a power-series inverse. Equivalently, every product $x g$ has constant coefficient $0$ .

givenF1

1.2

In $K ((x))$ , negative exponents are permitted and $x x^{- 1} = 1 = x^{- 1} x$ . Thus passing to Laurent series changes the answer.

givenF2

2.1

Steps 1.1 and 1.2 exhibit the claimed contrast.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Benjamin Sambale, An Invitation to Formal Power Series (standard reference, not scraped)

xx has no multiplicative inverse in R⟦x⟧R[[x]] although it is invertible in K((x))K((x)) when R=KR=K is a field