4. Let w = e, and let E = Q(w). (a) Verify that [E : Q] = 10. (b) Determine |Gal(E/Q)|, the number of automorphisms of E that fix Q. (c) Find the order of (w) = w² in Gal(E/Q). (d) How many subfields K does E have (such that QC KC E)? Hint 1 The corollary to Eisenstein's theorem might help. Hint 2: Example 6, page 535 may help.

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