Exercise 3, § 17

a) Let A be a central simple algebra of finite degree over K. Prove that the principal polynomial of an element $a$ of A is Pcrd$_{A/K}(a; X)$. The principal polynomial of A (with respect to a basis of A over K) is the polynomial P considered in Lemma 4 of VIII, p. 345 (reduce to the case $B =\mathbf{M}_n(K)$).

b) Let B be an absolutely semisimple algebra over K and $\mathscr{S}$ be the set of classes of simple B-modules. Prove that the principal polynomial of an element $b$ of B is equal to the product of the polynomials Pc$_{S/K}(b; X)$ for S running through $\mathscr{S}$ (reduce to the case when K is algebraically closed).