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Another simple fact about the Galois Group of a polynomial

Let $K/F$ be a Galois extension with Galois group $G$. For all $\alpha \in K$ we have the norm: $N_K(\alpha)=\alpha^2$.
It is given that $G$ is commutative and transitive, so why does the Galois Correspondence tell us that
$$N_K(a+b)=N_K(a) + N_K(b) \tag{$*$}$$
for all $a,b \in K$ with $a+b \in F$?
We only know it for $a,b \in K$ with $a+b \in F$ because we know that all elements of $K$ can be expressed as linear combinations of elements in $F$ (and all elements in $F$ are squares).
Do you have any hints? Thank you very much.

A:

I’ll write $\mathrm{Norm}(a)$ for $N_{K/F}(a)$.
For $1\leq i\leq j$ let $N_i\colon F\to F$ be the $i^\mathrm{th}$ coordinate. Then $\mathrm{Norm}(a)=\prod\limits_i N_i(a_i)$ where $a=\sum\limits_i a_i$.
The hypothesis $G=\mathrm{Aut}(K/F)$ tells us that the extension is Galois so that $K=F[\alpha]$ for some $\alpha\in K$ with $\alpha^2=N(\alpha)=\mathrm{Norm}(\alpha)$. It follows that the map $N_i$ induces maps $N_{i/F}\colon F\to F$ and $N_{i/K}\colon K\to K$ such that $N_{i/K}(\alpha)=N_i(\alpha)$.
In particular, for $0\leq k\leq l$ we get $N_{k/K}(\alpha)=\prod\limits_{i=k}^l N

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