Is every isogeny over $\mathbb Q$ separable?

Question

I am reading a proof of a simplified version of the weak Mordell-Weil theorem, where we only consider elliptic curves over $\mathbb Q$. Now, in the proof, they mention some (non-constant) isogeny, and it's clear from the context that they use that the isogeny is separable. I saw another source mention that any isogeny over $\mathbb Q$ is separable. Is this true, and could someone hint me in the direction as to why this would hold? (My knowledge on elliptic curves spans the first three chapters of Silverman's Arithmetic of Elliptic Curves, but I'm happy to be referred to other parts of the book.)

Answer 1

This has been answered in the comment, but let me expand it as an answer. By definition, an isogeny $f : E \to E'$ over a field $k$ (or more generally a non-constant morphism between two irreducible algebraic curves) is said to be separable if $f^*(k(E')) \subset k(E)$ is a separable extension of fields.

Now, if $k$ has characteristic $0$, e.g. $k = \Bbb Q$, then so is it the case for $k(E), k(E')$. Now any extension of a field $F$ of characteristic 0 must be separable: if $P \in F[X]$ is any irreducible polynomial, then $\gcd(P, P') = 1$ as it must divide $P$ and $P'$, and $\deg(P') = \deg(P) - 1$, so that $P$ is separable.