On the existence of finite Galois stable groups over integers in unramified extensions of number fields (original) (raw)

Publicationes Mathematicae Debrecen

Let E/F be a normal unramified number field extension with Galois group Γ 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.

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