At page 186 of the book "Joseph H. Silverman, John Tate - Rational points on elliptic curves-Springer-Verlag (1992)" there is a proposition where for any Galois extension $K/\mathbb{Q}$ it is defined the following action
$$Gal({K}/\mathbb{Q})\times E({K})\longrightarrow E({K})$$ $$(\sigma,(x,y))\longmapsto (\sigma(x),\sigma(y))$$ $$(\sigma,O)\longmapsto O$$ Later the author shows that it is an action and some good properties. But it doesn't seem necessary the condition of that the extension is Galois. My question is: is the Galois condition necessary?