The critical set of the reduced norm, as an algebraic set

For an associative algebra $A$ over $k$, the reduced norm $\nu :A \to k$ is a polynomial function. As such the critical set ${C_\nu }$ is defined, consisting of points where the differential ${(d\nu )_z}$ vanishes. The $k$-rational points of ${C_\nu }$ have been determined by the author under certain separability conditions (Rational critical points of the reduced norm of an algebra, Bull. Amer. Math. Soc. 80 (1974), 138-141); here the dimension of ${C_\nu }$ as an algebraic $k$-set is discussed.