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

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

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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