Definition
Global function field
A finite extension of a rational function field over a finite field.
Definition
A global function field is a finite field extension of for some finite field .
Equivalently, it is the function field of a smooth projective geometrically connected curve over a finite field, with the finite constant field understood as part of the global-field structure.
Places
Every place corresponds to a closed point of the associated projective curve and is nonarchimedean. Its completion is a finite extension of a Laurent-series field , hence a nonarchimedean local field of positive characteristic.
Position among global fields
Global function fields are precisely the global fields of positive characteristic. Their curves supply geometric methods that have no literal number-field counterpart.
References
- Michael Rosen, Number Theory in Function Fields, Springer, 2002, Chapters 1 and 5.
- Henning Stichtenoth, Algebraic Function Fields and Codes, second edition, Springer, 2009, Chapter I.