On the existence of finite Galois stable groups over integers in unramified extensions of number fields

Let E/F be a normal unramified number field extension with Galois group F of degree d, and let O_E be the ring of integers in E. It is proved that for given integers n, t such that n ≥ hφ_E(t)d, where h is the exponent of the class group of F and φ_E(t) is the generalized Euler function, there is a finite abelian γ-stable subgroup G ⊂ GL_n(O_E) of exponent t such that the matrix entries of all g ∈ G generate E over F. This result has certain arithmetic applications for totally real extensions, and a construction of totally real extensions having a prescribed Galois group is given.