Exercise 26, § 2

If E is an absolutely semi-simple algebra of finite rank over a commutative field k, the numerical invariant of E is the sequence of integers (n_1, ..., n_r) such that n_1 ≥ ... ≥ n_r > 0 and that \bar{k} ⊗_k E is isomorphic, for any algebraic closure \bar{k} of k, to $\prod_i M_{n_i}(\bar{k})$.

Let V be an integral ring, K its field of fractions, φ a homomorphism from V to a commutative field k, and E a V-algebra. Assume that E is a free V-module of finite rank and put E_0 = E ⊗_V k and E_1 = E ⊗_V K.

a) Assume that the bilinear form (x, y) ↦ Tr_{E_0/k}(xy) on E_0 is non-degenerate. Show that E_0 and E_1 are absolutely semi-simple (cf. Algebra, Chap IX, § 2, Exerc. 1).

b) Assume that E_0 and E_1 are absolutely semi-simple over k and K, respectively. Show that E_0 and E_1 have the same numerical invariant (reduce to the case where k and K are algebraically closed); let (e_i) be a basis of E over V and

let $(X_i)$ be indeterminates. Show that the characteristic polynomial of the element $\sum_i X_i e_i$ of $E_1 \otimes_K K[(X_i)]$ (resp. $E_0 \otimes_k k[(X_i)]$) is of the form $P = \prod_j P_j^{n_j}$ (resp. $Q = \prod_k Q_k^{m_k}$), where $(n_1, \ldots, n_r)$ (resp. $(m_1, \ldots, m_s)$) is the numerical invariant of $E_1$ (resp. $E_0$), with $\deg(P_j) = n_j$ (resp. $\deg(Q_k) = m_k$). Show that $P_j \in V[(X_i)]$ and that $Q = \varphi(P)$. Deduce that there exist integers $c_{jk} \geq 0$ such that