A simple proof of Abel’s theorem on the lemniscate

Since Abel’s original paper of 1827, his remarkable theorem on the constructibilityof the lemniscate splitting has been proven with the aid of Elliptic Functions. Nowadays, Rosen’s proof of 1981 is considered definitive. He also makesuse of (modern and more elaborate) Class Field Theory. Here we present anovel, short and simple proof of Abel’s Theorem on the lemniscate and itsconverse. Our only ingredients are the addition formulas of Gauss lemniscaticfunctions and some basic facts of Galois Theory