# Rendiconti del Seminario Matematico della Università di Padova

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**Volume 135, 2016, pp. 39–61**

**DOI: 10.4171/RSMUP/135-3**

Published online: 2016-05-19

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

Takeshi Takahashi^{[1]}(1) Niigata University, Japan

Let $C$ be a nonsingular plane quintic curve over the complex number field $\mathbb{C}$, and let $\pi_P\colon C \rightarrow \mathbb{P}^1$ be a projection from $P \in C$. Let $L_P$ be the Galois closure of the field extension $\mathbb{C}(C)/\mathbb{C}(\mathbb{P}^1)$ induced by $\pi_P$, where $\mathbb{C}(C)$ and $\mathbb{C}(\mathbb{P}^1)$ are the rational function fields of $C$ and $\mathbb{P}^1$, respectively. We call the point $P$ a $D_4$-point if the Galois group of $L_P/\mathbb{C}(\mathbb{P}^1)$ 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.

*Keywords: *Galois group of a projection, plane quintic curve, the dihedral group of order eight

Takahashi Takeshi: Projection of a nonsingular plane quintic curve and the dihedral group of order eight. *Rend. Sem. Mat. Univ. Padova* 135 (2016), 39-61. doi: 10.4171/RSMUP/135-3