Let $E/\mathbb{Q} = E\_{a,b}$
$$\displaystyle y^2 = x^3 + ax + b, a,b \in \mathbb{Z}$$
be an elliptic curve defined over the field of rational numbers, and let $n \geq 3$ be an integer. Let $K\_n$ be the $n$-torsion field of $E$, namely the field obtained by adjoining all of the coordinates of $n$-torsion points of $E$ to $\mathbb{Q}$. Let $p$ be a prime divisor of the discriminant $\Delta(E)$ of $E$. Does it follow that $K\_n$ is ramified at $p$? If so, what can be said about the splitting behaviour of $p$ in $K\_n$?



$n$-torsion fields of an elliptic curve defined over $\mathbb{Q}$
No. The modular curve $X\_E(n)$, which paramaterizes pairs $(E',\iota)$, where $E'$ is another elliptic curve and $\iota \colon E'[n] \to E[n]$ is an isomorphism (of group schemes, or equivalently, of Galois modules) is a twist of $X(n)$, and thus has genus 0 for $n = 2,3,4,5$ and genus 1 for $n = 6$. In particular, for $n = 3,4,5$, $X\_E(n)$ is isomorphic to $\mathbb{P}^1$ and has infinitely many rational points; i.e., there are infinitely many $E'/\mathbb{Q}$ such that $\mathbb{Q}(E'[n]) = \mathbb{Q}(E[n])$. But $\mathbb{Q}(E[n])$ is unramified outside of the primes dividing the conductor of $E$ or $n$, and most such $E'$ have bad reduction at new primes.


For primes greater than 6, the genus of $X(p)$ is at least 3. Anyway, this is related to the Frey--Mazur conjecture; but even if that conjecture is true, I don't think it implies what you're asking for $n > 6$. I do think what you are asking is true for large $n$ (independent of $E$), but as far as I know this is unknown.


