Exercise 110.22.7. Now suppose that $B$ is an $A$-algebra which is finite locally free as an $A$-module, in other words $B$ is a finite locally free $A$-algebra.

1. Define $\text{Trace}_{B/A}$ and $\text{Norm}_{B/A}$ using $\text{Trace}$ and $\det$ from Exercise 110.22.6.

2. Let $b\in B$ and let $\pi : \mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ be the induced morphism. Show that $\pi (V(b)) = V(\text{Norm}_{B/A}(b))$. (Recall that $V(f) = \{ {\mathfrak p} \mid f \in {\mathfrak p}\}$.)

3. (Base change.) Suppose that $i : A \to A'$ is a ring map. Set $B' = B \otimes _ A A'$. Indicate why $i(\text{Norm}_{B/A}(b))$ equals $\text{Norm}_{B'/A'}(b \otimes 1)$.

4. Compute $\text{Norm}_{B/A}(b)$ when $B = A \times A \times A \times \ldots \times A$ and $b = (a_1, \ldots , a_ n)$.

5. Compute the norm of $y-y^3$ under the finite flat map ${\mathbf Q}[x] \to {\mathbf Q}[y]$, $x \to y^ n$. (Hint: use the “base change” $A = {\mathbf Q}[x] \subset A' = {\mathbf Q}(\zeta _ n)(x^{1/n})$.)

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