Lemma 3

Let R be a commutative ring, $f(X) = X^d + \sum_{i=1}^d a_i X^{d-i}$ a monic polynomial of $\mathbf{R}[X]$, A the R-algebra $\mathbf{R}[X]/(f)$ and x the residue class of X in A. Then $\text{Tr}_{A/R}(x) = -a_1$ and $\text{N}_{A/R}(x) = (-1)^d a_d$.

By the Cor. (IV, p. 11) the sequence $(1, x, ..., x^{d-1})$ is a basis of A ; further we have

The matrix which expresses the multiplication by x with respect to the basis $(1, x, ..., x^{d-1})$ of $A$ is thus of the following form (we have taken $d = 5$ to fix the ideas):

The trace of this matrix is clearly $-a$; the determinant may be calculated by expanding by the first row, and we then find

For the rest of this No. we denote by E an extension of finite degree of K and by $x$ an element of E. We shall indicate how to calculate the norm and trace of $x$ in various cases.

a) Case of a separable extension: suppose that E is separable of degree $n$ over K, denote by $\Omega$ an algebraic closure of K and by $\sigma_1, \ldots, \sigma_n$ the $n$ distinct $K$-homomorphisms of E into R. By Formula (3) ($V$, p. 48) we have in R

b) Case of a p-radical extension: suppose that K is of characteristic $p > 0$ and that the extension E is p-radical; there exists an integer $e \geq 0$ such that $[E:K] = p^e$ ($V$, p. 26, Prop. 4). If $f$ is the height of $x$ over K, the minimal polynomial of $x$ over K is $X^{p^f} - x^{p^f}$ ($V$, p. 24, Prop. 1). By Prop. 3 we have $N_{E/K}(x) = (x^{p^f})^{p^e/p^f}$, whence

For the trace we find $\text{Tr}_{E/K}(x) = -p^{e-f} a$, where $a$ is the coefficient of $X^{p^f-1}$ in the polynomial $X^{p^f} - x^{p^f}$; in other words, we have

c) General case: we can summarize the calculation of norm and trace in the following proposition: