Proposition 9

Let V be a finite-dimensional right vector space over the field D. Let E be the algebra End$_D(V)$ over the field K; it is central simple and of finite degree. For every invertible element $u$ of E, we have

(64) Nrd$_{E/K}(u) =$ Nrd(det $u$)

(cf. VIII, p. 452, Proposition 2 for the definition of the determinant det $u$ of $u$).

We denote the dimension of V over D by $n$ and identify E with the matrix algebra $\mathbf{M}_n(D)$ using a basis of V over D. The multiplicative group GL$_n(D)$ of the algebra E is generated by the diagonal matrices and the matrices $B_{ij}(\lambda )$ (II, §10, No. 13, p. 362, Corollary 1). Proposition 9 therefore follows from the two specific cases below.

A) Suppose that $u$ is the diagonal matrix diag($a_1, . . . , a_n$). For every $1\leqslant i\leqslant n$, let $L_i$ be a maximal commutative subfield of D containing $a_i$; let L be the subalgebra of E consisting of the diagonal matrices diag($t_1, . . . , t_n$) with $t_i\in L_i$ for $1\leqslant i\leqslant n$. Let $d$ be the reduced degree of D over K. We have $[L_i: K] =d$ for $1\leqslant i\leqslant n$ (VIII, p. 265, Corollary 2). The K-algebra L is isomorphic to $L_1\times \cdots \times L_n$ and therefore semisimple of degree $nd$; now, we have $[E : K] =n^2[D : K] =n^2d^2= [L : K]^2$. It follows that L is a maximal commutative semisimple subalgebra of E (VIII, p. 262, Proposition 3). By Proposition 7 of VIII, p. 346, we have Nrd$_{E/K}(u) = N_{L/K}(u)$. Consequently, by formula (18) of III, §9, No. 3, p. 544, we have

Nrd$_{E/K}(u) =\prod_{i=1}^nN_{L_i/K}(a_i) =\prod_{i=1}^n$ Nrd$_{D/K}(a_i)$

= Nrd$_{D/K}(a_1\cdots a_n) =$ Nrd($\pi (a_1\cdots a_n)$).

Moreover, we have det $u=\pi (a_1\cdots a_n)$ by Proposition 3 of VIII, p. 453, which gives formula (64) in this case.

B) Suppose that $u$ is equal to $B_{ij}(\lambda )$, where $\lambda$ is an element of D and $i, j$ are distinct integers in the interval $[1, n]$. Denote by $d$ the reduced degree of D over K and by M the vector space over K deduced from V by restriction of scalars from D to K. Then M is a simple E-module, and we have

(65) Pc$_{M/K}(u; X) =$ Pcrd$_{E/K}(u; X)^d$

by formula (26) of VIII, p. 341. Moreover, M is a vector space of dimension $nd^2$ over K, and $u-1_M$ is a nilpotent endomorphism of M; we therefore have

(66) Pc$_{M/K}(u; X) = (X-1)^{nd^2}$.

By comparing formulas (65) and (66), we obtain

(67) Pcrd$_{E/K}(u; X) = (X-1)^{nd}$

and, in particular, Nrd$_{E/K}(u) = 1$. We also have det $u= 1$ by Proposition 3 of VIII, p. 453; formula (64) therefore holds in this case.