Remark 5

Suppose that $p \geq q$, and let $h \in A[X]$ be a polynomial of degree $\leq p - q$. Let us show that

(29) For if we write $\omega(u, v) = (u, uh + v)$ for $(u, v) \in S_q \times S_p$, then $w$ is an automorphism of the $A$-module $S_q \times S_p$ and we have The matrix of $w$ with respect to the basis $B_1$ is lower triangular and its diagonal elements are equal to $1$. On the other hand, $\varphi \circ w$ maps $(u, v)$ to $u(f + gh) + vg$. Now formula (29) means that the matrices representing $\varphi$ and $\varphi \circ w$ have the same determinant and this follows from the relation $\det w = 1$.

Assume that $f$ is monic of degree $p$; put $E = A[X]/(f)$ and denote by $x$ the canonical image of $X$ in $E$. We know (IV, p. 11) that $E$ is a free $A$-module with basis $(1, x, \ldots, x^{p-1})$. We can thus define the norm $N_{E/A}(u)$ of any element $u$ of $E$ (III, p. 543, Def. 2).