Remark

Let L be the field K(X) of rational fractions in one variable X. The reduced characteristic polynomial of an element $a$ of A is simply the reduced norm of the element $X\otimes 1-1\otimes a$ of the L-algebra $A_{(L)}$. This follows from the definition of the reduced characteristic polynomial and the formula (III, §9, No. 3, p. 544)

(25) Pc$_{A/K}(a; X) = N_{A_{(L)}/L}(X\otimes 1-1\otimes a)$.