Find the bound for [K(E[p]):K]

Question

Let E be an elliptic curve over a field K of characteristic p > 0, we know that E[p] has order 1 or p, how to bound [K(E[p]):K]?

Answer 1

The action of $\mathop{\mathrm{Gal}}(\overline{K}/K)$ on $E[p](\overline{K})$ gives a group homomorphism $$\mathop{\mathrm{Gal}}(\overline{K}/K)\longrightarrow \mathop{\mathrm{Aut}}(E[p](\overline{K}))\cong \begin{cases}0 \\ (\mathbf{Z}/p\mathbf{Z})^\times \end{cases}$$ The image of this homomorphism is (isomorphic to) $\mathop{\mathrm{Gal}}(K[E[p](\overline{K})]/K)$ and must have order dividing $p - 1$.