Abstract
Let $H$ be a division ring of finite dimension over its center, let $H[T]$ be the ring of polynomials in a central variable over $H$, and let $H(T)$ be its quotient skew field. We show that every intermediate division ring between $H$ and $H(T)$ is itself of the form $H(f)$ for some $f$ in the center of $H(T)$. This generalizes the classical Lüroth's theorem. More generally, we extend Igusa's theorem characterizing the transcendence degree 1 subfields of rational function fields, from fields to division rings.
Acknowledgments
We thank the anonymous referee for his/her comments, and in particular for providing us with Example 4.8. We thank Adam Chapman for his help with Lemma 2.2. This work fits into Project TIGANOCO (Théorie Inverse de GAlois NOn COmmutative), which is funded by the European Union within the framework of the Operational Programme ERDF/ESF 2014-2020.
Citation
Legrand François. Elad Paran. "Lüroth's and Igusa's theorems over division rings." Osaka J. Math. 61 (2) 261 - 274, April 2024.
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