Recently, in a study of Rational Points on Elliptic Curves, by Silverman and Tate, I went over a proof of Mordell's Theorem. The group of rational points on an elliptic curve is finitely generated. This was proved using the height function.
My question is: How simply can one prove that the group of complex points on an elliptic curve $E(\mathbb{C})$ is not finitely generated?
My proof starts like this:
Assume towards contradiction that the $E(\mathbb{C})$ is finitely generated. Then there exists a generating subset of $E(\mathbb{C})$. Denote this subset as $H$. Then my idea is that you can add complex points in the group and get the identity before generating the set, and this is the part that I am having trouble doing.
This is all I have so far... I don't know the best approach at deriving a contradiction.
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Any assistance would be greatly appreciated.
Finitely generated abelian groups are countable.