Show that group $Gal(K/\mathbb{Q})$ acts on $C(K)$ where $C$ is a rational elliptic curve

Question

Let $C$ be a rational elliptic curve, and let $K$ be a Galois extension of $\mathbb{Q}$.

a) Prove that for all $P \in C(K)$ and all $\sigma, \tau \in Gal(K/\mathbb{Q})$, $$ \tau(\sigma(P)) = (\tau\sigma)(P). $$ b) Prove that for all $P \in C(K)$ and all $\sigma \in Gal(K/\mathbb{Q})$, $$ \sigma(2P) = 2\sigma(P). $$

I know that any automorphism of $K$ has to send roots of $K$ to other roots of $K$, but I don't know how to show either part a or b. Any help would be much appreciated

Answer 1

Since $K:\mathbb{Q}$ is a galois extension then $σ(x)$ and $σ(y) \in K$. Therefore, for $P\neq \mathcal{O},$ we have $τ(σ(P))=τ(σ(x),σ(y))=(τ(σ(x)),τ(σ(y))=(τσ(x), τσ(y))= (τσ)(P)$