Let K be an infinite finitely generated field over its prime field and let A be an Abelian variety of dimension r over K. A theorem of Frey-Jarden from 1974 says that the rank of A(K_s(sigma)) is infinite for almost all sigma in Gal(K)^e. In the talk I'll prove an improvement of that theorem due to Geyer and myself: the rank of A(K_s[sigma]) is infinite for almost all sigma in Gal(K)^e. . Here Gal(K) := Gal(K_s/K) is the absolute Galois group of K equipped with the Haar measure. For each sigma=(sigma_1,...,sigma_e) in Gal(K)^e, K_s(sigma) denotes the fixed field in K_s of sigma_1,...,sigma_e, and K_s[sigma] denote the maximal Galois extension of K in K_s(\sigma).