Let K be a field, K[x] an irreducible polynomial of degree n 5 and
Question:
Let K be a field, ∫ ϵ K[x] an irreducible polynomial of degree n ≥ 5 and F a splitting field of ∫ over K. Assume that AutKF ≅ Sn. Let u be a root of ∫ in F. Then
(a) K(u) is not Galois over K; [K(u) : KJ = n and AutKK(u) = I (and hence is solvable).
(b) Every normal closure over K that contains u also contains an isomorphic copy of F.
(c) There is no radical extension field E of K such that E ⊃ K(u) ⊃ K.
Fantastic news! We've Found the answer you've been seeking!
Step by Step Answer:
Related Book For
Algebra Graduate Texts In Mathematics 73
ISBN: 9780387905181
8th Edition
Authors: Thomas W. Hungerford
Question Posted: