Trace and norm in towers - Imported Knowlpedia Content

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

\mathrm{Tr}_{L/K}(x)\in K,\qquad N_{L/K}(x)\in K

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

Theorem (tower formulas).

\mathrm{Tr}_{L/K}=\mathrm{Tr}_{E/K}\circ \mathrm{Tr}_{L/E},\qquad N_{L/K}=N_{E/K}\circ N_{L/E}.

Equivalent characterizations

Equivalently, for all $x\in L$ ,

\mathrm{Tr}_{L/K}(x)=\mathrm{Tr}_{E/K}\!\big(\mathrm{Tr}_{L/E}(x)\big),\quad N_{L/K}(x)=N_{E/K}\!\big(N_{L/E}(x)\big).

Remarks

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

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.

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 $\mathrm{Gal}(L/\mathbb{Q})$ .

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).