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I don't quite follow one piece of argument Lang uses about the norm and trace functions.

Given a finite field extension $E/k$, he defines the norm of $E/k$ s.t. $N(a)=\Pi_{i=1}^{r} \sigma_i(a^{p^u})$ where $r$ is the degree of separability and $p^u$ is the degree of inseparability of $E/k$. He then argues that $N(a)$ is invariant under the action of any embedding $\sigma_i$ of $E$ over $k$ and hence that $N(a)$ must lie in $k$ since $a^{p^u}$ is separable over $k$.

I don't understand how the separability of $a^{p^u}$ over $k$ comes into play here and why it forces $N(a)$ to lie in $k$.

EDIT: I can see that $N(a)$ is the product of the roots of the minimal polynomial of $a$ over $k$ and hence is a coefficient in that polynomial and hence lies in $k$. Similarly, the trace would be the sum of the roots. Is this right?

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    Possible duplicate of [Norm and trace proof in Lang](https://math.stackexchange.com/questions/2702537/norm-and-trace-proof-in-lang)2018-12-06

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