Lemma 9.20.5. Let $M/L/K$ be a tower of finite extensions of fields. Then

$\text{Trace}_{M/K} = \text{Trace}_{L/K} \circ \text{Trace}_{M/L} \quad \text{and}\quad \text{Norm}_{M/K} = \text{Norm}_{L/K} \circ \text{Norm}_{M/L}$

Proof. Think of $M$ as a vector space over $L$ and apply Lemma 9.20.4. $\square$

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