An irreducible elementary net over an algebraic extension that is not closed
For every order $n\geq3$ we construct an irreducible elementary net of modules over a principal ideal domain in an algebraic extension of its fraction field that is not closed. The example is defined over $\mathbb Z$ inside the cubic field generated by a root of $x^3-x-1$. This gives a negative answer to Kourovka Problem 19.48.
Lemma 2.1
The family $\sigma$ is an irreducible elementary net.
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- External referenceE. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, No. 21, Novosibirsk, 2026; arXiv:1401.0300.LaTeX source\bibitem{kourovka} E. I. Khukhro and V. D. Mazurov (eds.), \emph{Unsolved Problems in Group Theory: The Kourovka Notebook}, No. 21, Novosibirsk, 2026; arXiv:1401.0300.
- External referenceR. Steinberg, Lectures on Chevalley Groups, Yale University, 1968.LaTeX source\bibitem{steinberg} R. Steinberg, \emph{Lectures on Chevalley Groups}, Yale University, 1968.
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Machine-readable theorem index · 3 statements
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- PF:2026.000007/v1/LEM-2.1
The family $\sigma$ is an irreducible elementary net.
- PF:2026.000007/v1/THM-3.1
For every $n\geq3$, the elementary net $\sigma$ above is irreducible but not closed.
- PF:2026.000007/v1/COR-3.2
By Theorem 3.1, Kourovka Problem 19.48 has a negative answer, already for the PID $R=\Z$ and the cubic algebraic extension $K=\Q(\alpha)$.