Field Theory for Function Fields of Plane Quartic Curves

Let C be a smooth plane quartic curve over a field k and k(C) be a rational function field of C. We develop a field theory for k(C) in the following method. Let πP be the projection from C to a line l with a center P 2. The πP induces an extension field k(C)/k( 1), where k( 1) is a maximal rational subfield. In this paper we study the extension k(C)/k( 1) from several points of view. For example, we consider the following questions: When is the extension k(C)/k( 1) Galois? What is the Galois closure of k(C)/k( 1)?