Normal bases for quadratic extensions inside cyclotomic fields

摘要

If K is a number field, we write DK for the ring of integers in K. Let p be an odd prime, Cp a primitive p-th root of unity and L= Q(CP) the cyclotomic extension generated by ξp over Q. Several authors ([1], [2], [3], [6]) have investigated the Galois module structure of O_L over DN, where N is an intermediate field Q ∈N ∈L such that [L: N] = I. For example, Brinkhuis ([1], [2]) has shown that ξ>L has not a normal basis over ξN in case is an odd prime or l = 4. On the contrary, ξ>L has certainly a normal basis over DN when l = 2, that is, when N is the maximal real subfield of L; in this case (p generates a normal basis. This last fact seems to be well known ([5], p. 222). It is also very well known that Ox has a normal basis over % when K is the unique quadratic field contained in L; in this generates a normal basis.