# zbMATH — the first resource for mathematics

Function field theory of plane curves by dual curves. (English) Zbl 1064.14023
Summary: We study the structure of function fields of plane curves following our method developed previously in a joint paper with K. Miura [J. Algebra 226, 283–294 (2000; Zbl 0983.11067)]. Let $$K$$ be the function field of a smooth plane curve $$C$$ of degree $$d\geq 4$$ and let $$K_P$$ be a maximal rational subfield of $$K$$ for $$P\in \mathbb{P}^2$$. We study the field extension $$K/K_P$$ from a geometrical viewpoint. Especially, we give a sufficient condition that the Galois group of the Galois closure of $$K/K_P$$ becomes a full symmetric group.

##### MSC:
 14H05 Algebraic functions and function fields in algebraic geometry 12F10 Separable extensions, Galois theory 14H50 Plane and space curves
Full Text:
##### References:
 [1] Iitaka, S., Algebraic geometry, Graduate texts in math., 76, (1982), Springer-Verlag New York/Heidelberg/Berlin [2] Lang, S., Introduction to algebraic and abelian functions, Graduate texts in math., 89, (1995), Springer-Verlag New York/Heidelberg/Berlin [3] Miura, K.; Yoshihara, H., Field theory for function fields of plane quartic curves, J. algebra, 226, 283-294, (2000) · Zbl 0983.11067 [4] Miura, K.; Yoshihara, H., Field theory for the function field of the quintic Fermat curve, Comm. algebra, 28, 1979-1988, (2000) · Zbl 0978.14024 [5] Namba, M., Geometry of projective algebraic curves, (1984), Dekker New York/Basel · Zbl 0556.14012 [6] Namba, M., Branched coverings and algebraic functions, Research notes in math., 161, (1987), Pitman-Longman London [7] M. Oka, Geometry of cuspidal sextics and their dual curves, to appear. · Zbl 1020.14008 [8] Serre, J.P., Notes written by H. Darmon, “topics in Galois theory”, Research notes in math., 1, (1992), Jones and Bartlett Boston/London · Zbl 0746.12001 [9] Tokunaga, H., Triple coverings of algebraic surfaces according to the Cardano formula, J. math. Kyoto univ., 31, 359-375, (1991) · Zbl 0754.14010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.