All finite transcendence degree field extensions of " k " correspond to the rational function field of some variety.
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*PM : non-constant element of rational function field, id = 6762-- WP guess : non-constant element of rational function field-- Status:
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*PM : non-constant element of rational function field, id = 6762-- WP guess : non-constant element of rational function field-- Status:
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The definition by quadratic extensions of the rational function field works for fields in general except in characteristic 2; in all cases the geometric definition as a ramified double cover of the projective line is available, if it is assumed to be separable.
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In 1976, M . Rosen showed how to realize any countable abelian group as the class group of a Dedekind domain that is a subring of the rational function field of an elliptic curve, and conjectured that such an " elliptic " construction should be possible for a general abelian group ( Rosen 1976 ).
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