I am self-studying "The Arithmetic of Elliptic Curves", but I am having difficulties with Exercise 1.12(c). The problem is as follows:
Some definitions:
The "only if" part is easy. For the "if" part, I have made some attempts:
$\phi^\sigma=\phi$ implies that, for each $\sigma\in G_{\bar K/K}$ and $P\in V_1$,
$$[f_0^\sigma(P),\cdots,f_n^\sigma(P)]=[f_0(P),\cdots,f_n(P)].$$
Thus for a fixed $P$, and by (b), $[f_0(P),\cdots,f_n(P)]\in\mathbb P^n(K)$.
Then I got stuck here, and I couldn't figure out how to use (a). Any discussion will be greatly appreciated.
Write $\phi=[f_0,\ldots,f_n]$. Then, for every $\sigma \in G_K$, since $\phi^{\sigma}=\phi$, $f_i^{\sigma}/f_i=c_{\sigma}$ for some $c_{\sigma} \in Frac(\overline{K}(V_1))^{\times}$.
So, since $\phi$ is defined over some finite Galois extension $L/K$, we can in fact see $\sigma \longmapsto c_{\sigma}$ as an element of $H^1(Gal(L/K),Frac(L(V_1))^{\times})$.
So we can use Hilbert 90 if we can prove that $Frac(L(V_1))/Frac(K(V_1))$ is Galois of Galois group $Gal(L/K)$.
It’s enough to see that $Frac(L(V_1))^{Gal(L/K)}=K(V_1)$ (because no element of $Gal(L/K)$ acts trivially on $L(V_1)$).
The $\supset$ inclusion is trivial; so let $x=a/b \in Frac(L(V_1))^{Gal(L/K)}$. Let $B=\prod_{s \in Gal(L/K)}{b^s}$: then $B \in L(V_1)^{Gal(L/K)}=K(V_1)$ (by (a)), hence $a=xB \in L(V_1)^{Gal(L/K)}=K(V_1)$ (still by (a)), so $x \in Frac(K(V_1))$, which concludes.