Lemma 4

Let $u$ be the element $\sum_{i\in I}T_i\otimes e_i$ of the central simple $K(\mathbf{T})$-algebra $A_{(K(\mathbf{T}))}$. Let P be the reduced characteristic polynomial of $u$. Then P belongs to the ring $K[\mathbf{T}][X]$; viewed as an element of the ring $K[\mathbf{T},X]$, it is homogeneous of degree $n$.

We choose an extension L of K and an L-algebra isomorphism $\theta$ from $A_{(L)}$ to $\mathbf{M}_n(L)$. We denote by $\overline{\theta}: A_{(L(\mathbf{T}))}\rightarrow \mathbf{M}_n(L(\mathbf{T}))$ the isomorphism of $L(\mathbf{T})$-algebras deduced from $\theta$ by extension of scalars. By Corollary 1 of VIII, p. 342, we have

(47) $P(X) =\chi_{\overline{\theta}(u)}(X) =$ det(X$I_n-\overline{\theta}(u)$) $=$ det$(XI_n-\sum_{i\in I}T_i\theta (1\otimes e_i))$.

Since the matrices $\theta (1\otimes e_i)$ belong to $\mathbf{M}_n$(L), this formula shows that P is a homogeneous polynomial of degree $n$ in $L[\mathbf{T},X]$. It also belongs to $K(\mathbf{T})[X]$ and can be written as $P(X) =\sum_{j\geqslant 0}c_jX^j$, where each $c_j$ belongs to the intersection $K(\mathbf{T})\cap L[\mathbf{T}]$. By Lemma 3, each of the elements $c_j$ belongs to $K[\mathbf{T}]$; Lemma 4 follows.