Explicit Closed Elementary Nets That Cannot Be Completed, Including Fields of Transcendence Degree One

An elementary net (carpet) of order \(n\) over a field \(K\) is closed if its elementary net group contains no new elementary transvections, and it is completable if its diagonal can be supplemented to a full net. Completable nets are closed. Koibaev gave closed nets that are not completable over fields of characteristic \(0\) and \(2\) and asked for such nets in odd characteristic (Kourovka Notebook, Problem 21.76). Nuzhin (2026) has answered this question, with examples in every characteristic, using closures of finitely generated subgroups over a rational function field in two variables. We give explicit examples with short proofs. An elementary lifting lemma turns a closed pair of additive subgroups of a quotient ring \(R/J\) into a closed elementary net of every order \(n\ge 3\). With \(F[x,y]\to F[x]\) and a degree argument it gives, for every field \(F\), the net with \(Fx+yF[x,y]\) in positions \((1,2)\) and \((2,1)\) and \(yF[x,y]\) elsewhere. Lifting the two exceptional pairs of Levchuk’s refinement of Dickson’s theorem along \(\mathbb{F}_3[t]\to\mathbb{F}_9\) and \(\mathbb{F}_2[t]\to\mathbb{F}_4\) gives closed nets over \(\mathbb{F}_3(t)\) and \(\mathbb{F}_2(t)\) that are not completable. These one-variable nets also satisfy the hypotheses of Kourovka Problem 19.48. Together with a theorem of Koibaev and Nuzhin on algebraic extensions, it follows that in characteristics \(2\) and \(3\) a field carries such nets if and only if it is not algebraic over its prime field. For \(p\ge 5\) we do not know whether examples exist over \(\mathbb{F}_p(t)\); by Levchuk’s theorem, lifting from a finite field cannot produce them. This is an unrefereed note.