primitive pth root of unity and cyclotomic fields
Let n = p be an odd prime number and ζ be a primitive pth root of unity. Show that Q(ζ) contains Q(√p) if p≡1 mod 4, and contains Q(√−p) if p≡3 mod 4.
First, define an element S ∈ Q(ζ) by
S = SUM[(a/p)*(ζ^a)] over a, where a is an element in (Z/pZ)*, and (a/p) is the quadratic residue symbol.
Is there a cyclotomic field containing both p^(1/2) and (-p)^(1/2)?
Is there a cyclotomic field containing
(iii) both 2^(1/2) and (-2)^(1/2)
(iv) m^(1/2) for any integer m, positive or negative?
(In each case, give a constructive proof or prove impossibility)