This is question from Silverman's 'The arithmetic of elliptic curves', p70.
Let $E:y^2=x^3-x/K$, then $[i]:(x,y)→(-x,iy)$ is endomorphism of $E$. The book reads ring hom $ \Bbb{Z}[i]→End(E), m＋ni→[m]＋[n]・[i]$ is isomorphism if $char(K)=0$.
But Why does this hold ? The case $K= \Bbb{C}$ is well discussed in chapter Ⅵ, but in general field, how can I prove this ?
In Section 9 of chapter 3, Silverman proves that $\text{End}(E)$ is one of the following three types of rings:
1.) $\mathbb{Z}$
2.) An order in an imaginary quadratic extension of $\mathbb{Q}$
3.) An order in a quaternion algebra over $\mathbb{Q}$
and that furthermore if $\text{char}(K) = 0$, then only the first two are possible. It is not hard to show that the homomorphism of $\mathbb{Z}[i]$ into $\text{End}(E)$ Silverman gives on page 70 is injective, hence $\text{End}(E)$ must be an order in an imaginary quadratic extension of $\mathbb{Q}$ that contains $\mathbb{Z}[i]$. Since quadratic extensions of $\mathbb{Q}$ are pairwise non-isomorphic, it follows that $\text{End}(E) \cong \mathbb{Z}[i]$.