(8) Let f(x) ∈ Z[x] be an irreducible polynomial of degree 4 such that its Galois group over Q is isomorphic to S4. Let α be a root of f(x). Show that Q(α) has no subfields other than Q and Q(α).
(8) Let f(x) ∈ Z[x] be an irreducible polynomial of degree 4 such that its Galois group over Q is isomorphic to S4. Let α be a root of f(x). Show that Q(α) has no subfields other than Q and Q(α).
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter7: Eigenvalues And Eigenvectors
Section7.CM: Cumulative Review
Problem 16CM
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(8) Let f(x) ∈ Z[x] be an irreducible polynomial of degree 4 such that its Galois group over Q is isomorphic to S4. Let α be a root of f(x). Show that Q(α) has no subfields other than Q and Q(α).
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