TITLE: How do you see the genus of a curve, just looking at its function field?
QUESTION [26 upvotes]: Yuhao asked in the 20-questions seminar:
The genus of a curve is a birational invariant; the function field of a curve determines it up to birational equivelance.
How do you see the genus directly from the field?

REPLY [2 votes]: See C. Chevalley, Introduction to the theory of Algebraic functions of one variable. That book develops the whole theory of algebraic curves using just its function field.