Lemma 3

Let L be an extension of K, and let I be a set and $\mathbf{T}= (T_i)_{i\in I}$ a family of variables. We have $K(\mathbf{T})\cap L[\mathbf{T}] = K[\mathbf{T}]$.

Let P and Q be elements of $K[\mathbf{T}]$ with $Q\not= 0$. The coefficients of the polynomials R in $L[\mathbf{T}]$ such that P = QR are the solutions of a system of linear equations with coefficients in K. Consequently, if there exists a polynomial $R\in L[\mathbf{T}]$ such that P = QR, then there also exists one in $K[\mathbf{T}]$ (II, §8, No. 5, p. 321, Proposition 6). This proves the inclusion $K(\mathbf{T})\cap L[\mathbf{T}]\subset K[\mathbf{T}]$; the reverse inclusion is obvious.

Recall that the reduced characteristic polynomial of an element $a$ of A can be written as

(46) Pcrd$_{A/K}(a; X) =\sum_{r=0}^n(-1)^rb_r(a) X^{n-r}$

and that we have

$b_0(a) = 1,b_1(a) =$ Trd$_{A/K}(a),b_n(a) =$ Nrd$_{A/K}(a)$.