Monogenity of totally real algebraic extension fields over a cyclotomic field

Let K   be a composite field of a cyclotomic field knkn of odd conductor n≧3n≧3 or even one ≧8 with 4|n4|n and a totally real algebraic extension field F over the rationals Q   and both fields knkn and F are linearly disjoint over Q to each other. Then the purpose of this paper is to prove that such a relatively totally real extension field K   over a cyclotomic field knkn has no power integral basis. Each of the composite fields K   is also a CM field over the maximal real subfield kn+⋅F of K  . This result involves the previous work for K=kn⋅FK=kn⋅F of the Eisenstein field kn=k3kn=k3 and the maximal real subfields F=kpn+ of prime power conductor pnpn with p≧5p≧5, and an analogue K=kn⋅FK=kn⋅F of cyclotomic fields kn=k2m(m≧3) with a totally real algebraic fields F   of K=k4⋅FK=k4⋅F with a cyclic cubic field F   except for k4⋅k7+ and k4⋅k32+ of conductors 28 and 36.