# Projection of a nonsingular plane quintic curve and the dihedral group of order eight

### Takeshi Takahashi

Niigata University, Japan

## Abstract

Let $C$ be a nonsingular plane quintic curve over the complex number field $C$, and let $π_{P}:C→P_{1}$ be a projection from $P∈C$. Let $L_{P}$ be the Galois closure of the field extension $C(C)/C(P_{1})$ induced by $π_{P}$, where $C(C)$ and $C(P_{1})$ are the rational function fields of $C$ and $P_{1}$, respectively. We call the point $P$ a $D_{4}$-point if the Galois group of $L_{P}/C(P_{1})$ is isomorphic to the dihedral group $D_{4}$ of order eight. In this paper, we prove that the number of $D_{4}$-points for $C$ equals $0$, $1$, $3$, $5$, or $15$, and show that the curve with $15$ $D_{4}$-points is projectively equivalent to the Fermat quintic curve.

## Cite this article

Takeshi Takahashi, Projection of a nonsingular plane quintic curve and the dihedral group of order eight. Rend. Sem. Mat. Univ. Padova 135 (2016), pp. 39–61

DOI 10.4171/RSMUP/135-3