ABSTRACT
The determination of the genus proceeds in steps: write your function field F as an algebraic extension of a function field E ⊂ F whose genus is known. The Riemann-Hurwitz formula relates the canonical divisor of F to that of E and therefore 2g(F) − 2 to 2g(E) − 2. A typical application of this method uses E = K(x), a rational function field. It follows that we need to study the relationship between the basic notions (places, divisors, residue class fields, genus, etc.) of E and of F ⊃ E.
