Solitary Galois Extensions of Algebraic Number Fields Original Research Article

A finite extension K of an algebraic number field k is called k-solitary if for any finite extension L of k the equality of norm groups N K/k K* = N L/k L* implies that K and L are conjugate over k (i.e., K and L are k-isomorphic). In the present work we investigate finite Galois extensions of k which are k-solitary. We prove that if a finite Galois extension K of k is k-solitary, then K/k is a 2-extension. k and every quadratic extension of k is k-solitary. We establish a necessary condition for a finite Galois 2-extension K/k with (K : k) ≥ 4 to be k-solitary. We then investigate covering subgroups, and show that the necessary condition is also a sufficient condition for extensions of degree 4 or 8. We conclude our discussion by constructing a Galois extension E of the field of rational numbers Q with the Galois group isomorphic to the dihedral group of order 8 so that E is Q-solitary.