Trace and Norm in Towers

Theorem (Tower formulas).
Let $K \subseteq L \subseteq M$ be a tower of finite extensions. Then:

• (Trace) $\mathrm{Tr}_{M/K} = \mathrm{Tr}_{L/K}\circ \mathrm{Tr}_{M/L}$ as maps $M\to K$.
• (Norm) $N_{M/K} = N_{L/K}\circ N_{M/L}$ as maps $M^\times\to K^\times$.

These are the standard trace and norm compatibility relations, and they pair naturally with the tower law for degrees.

Examples.

1. If $K\subseteq L\subseteq M$ and $\alpha\in M$, then $\mathrm{Tr}_{M/K}(\alpha)=\mathrm{Tr}_{L/K}(\mathrm{Tr}_{M/L}(\alpha))$.
2. For finite fields $\mathbb{F}_q \subseteq \mathbb{F}_{q^m} \subseteq \mathbb{F}_{q^{mn}}$, norms satisfy $N_{\mathbb{F}_{q^{mn}}/\mathbb{F}_q} = N_{\mathbb{F}_{q^m}/\mathbb{F}_q}\circ N_{\mathbb{F}_{q^{mn}}/\mathbb{F}_{q^m}}$.
3. For quadratic towers $K \subset L \subset M$ with both steps degree $2$, norms multiply: $N_{M/K}(\alpha)=N_{L/K}(N_{M/L}(\alpha))$.

Related. discriminant , trace/norm tower law .