Birational classification of pointless del Pezzo surfaces of degree 8

Let k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Bbbk $$\end{document} be a perfect field. Recently Jean-Louis Colliot-Thélène showed that two pointless quadric surfaces over k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Bbbk $$\end{document} are birationally equivalent if and only if they are isomorphic. We show that this result holds for arbitrary del Pezzo surfaces of degree 8 with the Picard number 1, and describe minimal surfaces birationally equivalent to a given pointless del Pezzo surface of degree 8.