function field (original) (raw)

Definition 1.

Definition 3.

Let K be a function field over F and let L be a finite extension of K. The extension L/K of function fields is said to be geometric if L∩F¯=F.

Example 1.

The extension ℚ⁢(x)/ℚ⁢(x) is geometric, but ℚ⁢(2)⁢(x)/ℚ⁢(x) is not geometric.

Theorem 1 (Thm. I.6.9 of [1]).

Let K be a function field over an algebraically closed field F. There exists a nonsingular projective curve CK such that the function field of CK is isomorphic to K.

Definition 4.

Let K be a function field over a field F. Let K′=K⁢F¯ which is a function field over F¯, a fixed algebraic closure of F, and let CK′ be the curve given by the previous theorem. The genus of K is, by definition, the genus of CK′.

References