LÜROTH’S AND IGUSA’S THEOREMS OVER DIVISION RINGS

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, andletH(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)forsome 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.