We study the structure of function fields of plane curves following our method Ž . developed previously K. Miura and H. Yoshihara, 2000, J. Algebra 226, 283᎐294 .Ž . Let K be the function field of a smooth plane curve C of degree d G 4 and let K be a maximal rational subfield of K for P g ‫ސ‬ 2 . We study the field extension P KrK from a geometrical viewpoint. Especially, we give a sufficient condition that P the Galois group of the Galois closure of KrK becomes a full symmetric group.

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 π P be the projection from C to a line l with a center P ∈ 2 . The π P induces an extension field k C /k 1 , where k

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