Let $K$ be a finite extension of $\mathbb{Q}$, and let $Hom(K,\mathbb{C})$ be the set of $\mathbb{Q}-$homomorphisms from $K$ to $\mathbb{C}$.
Define the norm of $a\in K$ to be $$N(a)=\prod_{\sigma \in Hom(K,\mathbb{C})}\sigma(a)$$
And we want to prove that $N(a)$ is a rational number.
I saw a proof somewhere but I couldn't understand it well:
A possible proof
Suppose $\tau\in Gal(\mathbb{C}/\mathbb{Q})$
Since by replacing every $\sigma(a)$ by $\tau\sigma(a)$ we get a permutation of the product in the definition of $N(a)$, so we can easily have $$\tau(N(a))=\prod_{\sigma}\tau\sigma(a)=N(a)$$
By this we have $N(a)\in \mathbb{Q}$
My confusion about this proof
I have no idea about the fixed field of the galois group $Gal(\mathbb{C}/\mathbb{Q})$, so I don't know why we can deduce $N(a)\in \mathbb{Q}$ from the above equation.
My question is: why is the above "proof" valid or not?
Thank you very much!