On higher trace forms of separable algebras

Let k be a field and A be a separable k-algebra. Let r ≥ 3 be an integer. Generalizing a result of Reichstein (Arch. Math. 88 (2007), 12–18), we prove that the essential dimension over k of A equals that of the r-fold trace form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_r: (a_1,\ldots,a_r)\mapsto \mbox{tr}(a_1\ldots a_r)$$\end{document}.