Given an elliptic curve $E$ there is the multiplication by $n$ map $[n]: E \rightarrow E$.


If $K(E)$ is the fraction field then this map makes $K(E)$ a degree $n^2$ extension of itself. 


My question is what are some techniques for dealing with the field theory of this extension? Can one write down generators for this extension in terms of the coefficients of E?


If not, what are popular techniques for dealing with the Galois theory of this extension?



Elliptic curve multiplication on the generic point
If $K$ is an algebraically closed field whose characteristic doesn't divide $n$, then the Galois theory of this extension is not complicated. Indeed $K(E)/[n]^\* K(E)$ is then abelian with Galois group canonically isomorphic to $E[n]$. A point $P \in E[n]$ acts by translation on $K(E)$, namely $\sigma\_P(f) = t\_P^\* f$ where $t\_P : E \to E$ is the translation map, and the $\sigma\_P$ are exactly the elements of the Galois group. A function $f \in K(E)$ will generate the extension if and only if its translates by $E[n]$ are pairwise distinct. It is not hard to show that, taking for example a Weierstrass model for $E$, the function $f=x$ works.


In the case $K$ is not algebraically closed one has to be careful that the extension isn't necessarily Galois anymore (it is if and only if the $n$-torsion is rational over $K$). However, regarding the question about generators, the above criterion generalizes, namely $f \in K(E)$ generates the extension if and only if its translates (which lie in $K\_n(E)$, where $K\_n$ is the field obtained from $K$ by adjoining the coordinates of the $n$-torsion points) are pairwise distinct. So, for example, $f=x$ is still a generator of the extension.


**EDIT.** In view of the OP's comments, here is an alternate way of constructing generators of the extension $K(E)/[n]^\* K(E)$. These generators will have minimal polynomials of the form $X^n-[n]^\* g$, for reasonable functions $g \in K(E)$.


Assume $E[n] \subset E(K)$, so that $K(E)/[n]^\* K(E)$ is Galois and $\operatorname{Gal}(K(E)/[n]^\* K(E)) \cong E[n]$, as explained above. By Kummer theory, this extension is generated by two $n$-th roots of suitable elements of $[n]^\* K(E)$. We can find these elements as follows. Let $f \in K(E)$ such that $f^n = [n]^\* g$ for some $g \in K(E)$. Taking the divisors, we see that $\operatorname{div}(f)$ is invariant by translation by $E[n]$, so it has the form


\begin{equation\*}
\operatorname{div}(f) = \sum\_i \sum\_{R \in E[n]} \lambda\_i [P\_i+R]
\end{equation\*}
with $\lambda\_i \in \mathbf{Z}$ and $P\_i \in E$. Conversely, such a divisor is principal if and only if $\sum \lambda\_i = 0$ and $n^2 \sum \lambda\_i P\_i = 0$ (because the sum of all $n$-torsion points is zero). The divisor of $g$ is then given by


\begin{equation\*}
\operatorname{div}(g) = n \sum\_i \lambda\_i [n P\_i]
\end{equation\*}
Note that if $n \sum \lambda\_i P\_i = 0$ then $f$ is already in $[n]^\* K(E)$. So, letting $Q\_i=nP\_i$, the conditions on the divisor of $g$ are $\sum \lambda\_i = 0$ and $\sum \lambda\_i Q\_i \in E[n] \backslash \{0\}$.


For example, for each $Q \in E[n]$, one can take $g\_Q$ such that $\operatorname{div}(g\_Q)=n[Q]-n[0]$. Identifying the base field of the extension with $K(E)$ and denoting the extension by $L/K(E)$, we thus get $L=K(E)(g\_{Q\_1}^{1/n},g\_{Q\_2}^{1/n})$ if and only if $(Q\_1,Q\_2)$ is a basis of $E[n]$. Note : the Galois action on these functions $g\_{Q}^{1/n}$ is related to the Weil pairing (see for example Silverman, *The arithmetic of elliptic curves*). More generally one can define $g\_D$ for any $D \in I$, where $I$ is the augmentation ideal of $\mathbf{Z}[E[n]]$. Then $L=K(E)(g\_{D\_1}^{1/n},g\_{D\_2}^{1/n})$ if and only if the classes of $D\_1$ and $D\_2$ generate $I/I^2 \cong E[n]$. Finally, there is the further flexibility in that one can choose functions $g$ whose divisors are not necessarily supported on $E[n]$.


