In the following, $\wp$ denotes the Weierstrass elliptic function (https://en.wikipedia.org/wiki/Weierstrass_elliptic_function), $g_2$ and $g_3$ are invariants and $\omega_1$ and $\omega_2$ are periods.
Page 11 of this presentation (http://swc.math.arizona.edu/aws/2008/08WaldschmidtSlides3.pdf) states
(C. L. Siegel, 1932): Assume the invariants $g_2$ and $g_3$ of $\wp$ are algebraic. Then at least one of $\omega_1,\omega_2$ is transcendental.
Where can I find a proof of this theorem? The presentation doesn't include a list of references.
You can find the proof in Siegel, C.L., "Über die Perioden elliptischer Funktionen", Journal für die reine und angewandte Mathematik 167 (1932): 62-69. http://eudml.org/doc/149791.