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

Fr: (a1,?,ar)? tr(a1?ar)F_r: (a_1,\ldots,a_r)\mapsto \mbox{tr}(a_1\ldots a_r)
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