Talk:Minimal splitting field need not be contained in a cyclotomic extension of rationals

Notes:

Note, however, that since sufficiently large implies splitting, any minimal splitting field must be contained in a cyclotomic extension of the rationals. This is FALSE. There are minimal splitting fields for Q8 which do not lie in any cyclotomicf field. One such is Q(b,a) = Q(b), where b^2 = -1-a^2 and a^3 = 2. Only proper subfields, Q and Q(a), have real embeddings. Seems one can allow a^n = 2 for n odd, n \ne 5. [If n = 5, then Q(b) contains Q(\sqrt{-5}).] Note also that the degree [6] of this field (over Q) differs from the Schur index [2] (over Q) of the only abs irred representation of Q8 not realisable over Q. [NB: There are some imaginary quadratic fields over which Q8 cannot be realised, for example Q(\sqrt{-7}). For square-free d > 0, Q8 can be realised over Q(\sqrt{-d}) precisely when d \not\equiv 7 (mod 8).]