Unformatted text preview: that K ⊃ Q ( e 2 πi/N ). For each such N , ﬁnd the degree [ K : Q ( e 2 πi/N )]. 4. (15 points) Let ζ = e 2 πi/ 15 ∈ C and let K = Q ( ζ ). Show that [ K : K ∩ R ] = 2. 5. (15 points) Let ζ = e 2 πi/ 15 . Show that (1 + 2 ζ 3 + 2 ζ 12 ) 2 ∈ Q . Conclude that [ Q (1 + 2 ζ 3 + 2 ζ 12 ) : Q ] = 2. Hint: Cyclotomic polynomials may help. 6. (25 points) Let ζ = e 2 πi/ 15 and let K = Q ( ζ ). (a) Show that K is a Galois extension of Q , and ﬁnd the Galois group G ( K : Q ). (b) Find all subgroups of G ( K : Q ), and the inclusion relations among them. (c) Exhibit as many intermediate ﬁelds for the extension K ⊃ Q as you can. (Hint: Use earlier problems on this exam.) (d) Be as explicit as you can about the Galois correspondence between subgroups of the Galois group G ( K : Q ) and intermediate ﬁelds for the extension K ⊃ Q ....
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 Spring '08
 MORRISON
 Math, Algebra, Galois theory, Galois group, primitive 15th root

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