In "M. Kaneko and D. Zagier, A generalized Jacobi theta function and quasimodular forms, Prog. Math. 129,
165-172 (1995)" there is a proposition stating essentially that $E\_2$, $E\_4$ and $E\_6$ are algebraically independent.


Unfortunately there is no proof there. I know how to show the algebraic independence of $E\_4$ and $E\_6$.


How does the proof for $E\_2$ work?



Algebraic independence of $E_2$, $E_4$ and $E_6$
Here is a reference for the algebraic independance of $E\_2,E\_4,E\_6$ over $\mathbf{C}$ :


[MR2186573 (2007a:11065) Martin, François ; Royer, Emmanuel . Formes modulaires et périodes.
(French) [Modular forms and periods] Formes modulaires et transcendance, 
 1--117, Sémin. Congr., 12, Soc. Math. France, Paris, 2005.](http://smf4.emath.fr/Publications/SeminairesCongres/2005/12/pdf/smf_sem-cong_12_1-117.pdf)


See Lemme 117 p. 80. The proof is along the lines suggested by Emerton.


