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Unformatted text preview: that K Q ( e 2 i/N ). For each such N , nd the degree [ K : Q ( e 2 i/N )]. 4. (15 points) Let = e 2 i/ 15 C and let K = Q ( ). Show that [ K : K R ] = 2. 5. (15 points) Let = e 2 i/ 15 . Show that (1 + 2 3 + 2 12 ) 2 Q . Conclude that [ Q (1 + 2 3 + 2 12 ) : Q ] = 2. Hint: Cyclotomic polynomials may help. 6. (25 points) Let = e 2 i/ 15 and let K = Q ( ). (a) Show that K is a Galois extension of Q , and nd 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 elds 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 elds for the extension K Q ....
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 Spring '08
 MORRISON
 Math, Algebra

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