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arXiv · 2304.10738

L\"uroth's and Igusa's theorems over Division Rings

François Legrand·Elad Paran
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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 L\"uroth's theorem. More generally, we extend Igusa's theorem characterizing the transcendence degree 1 subfields of rational function fields, from fields to division rings.

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François Legrand, Elad Paran. 2023-04-21. L\"uroth's and Igusa's theorems over Division Rings. https://arxiv.org/abs/2304.10738

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