On the structure of elliptic fields. I

Let $\mathscr K$ be an algebraic field and $\mathscr F$ an elliptic curve defined over $\mathscr K$. Let $\{O_{p^t},O'_{pt}\}$ be a basis of all the points of order $p^t$ on $\mathscr F$. The field $\mathscr K(O_{p^t},O'_{pt})/\mathscr K(O_p)$ is given explicitly.
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