About Mordell's Theorem (Elliptic Curves) I've just finished the proof of Mordell's Theorem given in the book "Rational Points on Elliptic Curves " by Silverman. One of the key lemmas used in the proof of the theorem is: Let $C(\mathbb{Q})$ denote the group of rational points of $C$ then $[C(\mathbb{Q}):2C(\mathbb{Q})]$ is finite. But in the book the lemma is proved under an additional assumption saying that $C$ has a rational point of order 2. I'd like to know how much algebraic number theory is needed to avoid that assumption and some references to see if I could try to look at a more general proof.

The proof of that result, usually called (an explicit version of) the Weak Mordell-Weil theorem, can be found in Silverman's _Arithmetic of Elliptic Curves_ book.

The proof uses Galois cohomology and some minor arithmetic that can be followed with the knowledge of a few of the main theorems of global class field theory.