## Rational Function Field

The rational function field *F* = **F**_{b}(*x*) is an algebraic function field with full constant field **F**_{b} and genus *g*(*F*/**F**_{b}) = 0. It has *N*(*F*/**F**_{b}) = *b* + 1 rational places, which can be identified with the *b* zeros of the linear polynomials and with the pole of *x*, usually denoted by ∞. The corresponding curve is the projective line PG(1, *b*), the rational points are *P*_{∞} = (1 : 0) and *P*_{x} = (*x* : 1) for *x* ∈ **F**_{b}.

### Optimality

Rational function fields meet the upper Hasse-Weil-Bound with equality and are therefore maximal function fields.

### Copyright

Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.

Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “Rational Function Field.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_FRational.html