Let $K$ be a number field and let $E$ be an elliptic curve over $K$ . By $E(K)$ we denote the set of points in $E$ with coordinates in $K$ .

Proof. The proof of this theorem is fairly involved. The main two ingredients are the so called weak Mordell-Weil theorem, the concept of height function for abelian groups and the ``descent'' theorem.
See $\cite{silverman}$ , Chapter VIII, page 189.

Bibliography

1

James Milne, Elliptic Curves, online course notes. http://www.jmilne.org/math/CourseNotes/math679.html

2

Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.

3

Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1994.

4

Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton, New Jersey, 1971.