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midterm-round2

midterm-round2 - that K ⊃ Q e 2 πi/N For each such N...

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Math 220C Name Spring 2008 Midterm Exam – Round Two May 5, 2008 Instructions: (1) In round two of the midterm, you may supplement the answers you gave during the in-class exam in any way you choose. This is an open-book exam, but you may not consult other people. “Round two” is due at 6 p.m. on Wednesday, May 7 in room 6635 South Hall (at the start of the makeup lecture). (2) There are six problems, and 100 total points, on this exam. 1. (15 points) Let L be a subfield of the complex numbers C such that [ L : Q ] = 2. Let α Q such that α Q and Q ( α ) = L . Let M be the smallest subfield of C containing both Q ( α ) and L . Show that [ M : Q ] = 4. How many intermediate subfields are there? 2. (15 points) Let ζ = e 2 πi/ 15 and let K = Q ( ζ ). Show that any automorphism of the field K must map ζ to a primitive 15 th root of unity, and that every primitive 15 th root of unity is the image of ζ under at least one automorphism of K . 3. (15 points) Let ζ = e 2 πi/ 15 and let K = Q ( ζ ). Find all natural numbers N such
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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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