Trace and norm in towers

Let $K\subseteq E\subseteq L$ be a tower of fields of finite extensions. For $x\in L$, write

for the field trace and field norm . Then trace and norm are compatible with towers:

Theorem (tower formulas).

Equivalently, for all $x\in L$,

These identities are compatible with degree calculations from the tower law and reflect the fact that trace and norm can be defined as the trace/determinant of the $K$-linear map “multiplication by $x$” on $L$.

Examples

1. A quadratic subextension.
   Take $K=\mathbb{Q}$, $E=\mathbb{Q}(\sqrt2)$, $L=E(\sqrt3)=\mathbb{Q}(\sqrt2,\sqrt3)$. For $y=a+b\sqrt2\in E$, $\mathrm{Tr}_{E/K}(y)=2a$ and $N_{E/K}(y)=a^2-2b^2$. The tower formulas then compute $\mathrm{Tr}_{L/K}$ and $N_{L/K}$ by first taking $\mathrm{Tr}_{L/E}$, $N_{L/E}$ and then applying the quadratic formulas above.

2. Cyclotomic example.
   With $K=\mathbb{Q}\subseteq E=\mathbb{Q}(\zeta_3)\subseteq L=\mathbb{Q}(\zeta_9)$, the tower identities express $\mathrm{Tr}_{L/\mathbb{Q}}$ and $N_{L/\mathbb{Q}}$ in terms of intermediate traces/norms, often simplifying computations because $\mathrm{Gal}(L/E)$ is smaller than $\mathrm{Gal}(L/\mathbb{Q})$.

3. Finite fields.
   For $\mathbb{F}_p \subseteq \mathbb{F}_{p^m}\subseteq \mathbb{F}_{p^n}$ with $m\mid n$, the trace and norm are given by explicit Frobenius sums/products, and the tower formulas reflect the composition of these Frobenius patterns (see finite-field cyclic Galois group ).