# Question about Q(cos(2π /n))

The following question is related to Q(cos(2π /n)) and Q(ζn) where ζn is the n th root of unity.
(a) Using the fact that 2cos(2π /n)= ζn +ζn-1 , or otherwise, show that all the numbers cos(2π /n), cos(4π /n), cos(6π /n)… belong to the field Q(cos(2π /n))
(b)...
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The following question is related to Q(cos(2π /n)) and Q(ζn) where ζn is the n th root of unity.

(a) Using the fact that 2cos(2π /n)= ζn +ζn-1 , or otherwise, show that all the numbers cos(2π /n), cos(4π /n), cos(6π /n)… belong to the field Q(cos(2π /n))

(b) Show that (Q(ζn): Q(cos(2π /n)))=2 for n>2, and hence show that the degree of cos(2π /n) is φ(n)/2. Use this to prove that cos(2π /7) and cos(π /9) has degree 3

(c) For which integer n is cos(2π /n) rational

(d) Use (b) to show that cos(2π /n) is of degree 3 only for n=7,9,14,18. Also find the n for which cos(2π /n) is of degree 4.

(e) Show that any isomorphisms of Q(cos(2π /n) are isomorphisms of Q(ζn).

(f) Show that for each k relatively prime to n there is an automorphism σk of Q(cos(2π /n) defined by σk(cos(2π /n))=cos(2kπ /n), and that every automorphisms of Q(cos(2π /n) is of this form.(compared with (e))

(g) How many of the automorphisms σk of Q(cos(2π /n) are actually distinct? Compare this number with the value of (Q(cos(2π /n) : Q) found in (b)

(h) Use (g) or otherwise, find fields Q(cos(2π /n) with automorphism groups of three and five elements.

(a) Using the fact that 2cos(2π /n)= ζn +ζn-1 , or otherwise, show that all the numbers cos(2π /n), cos(4π /n), cos(6π /n)… belong to the field Q(cos(2π /n))

(b) Show that (Q(ζn): Q(cos(2π /n)))=2 for n>2, and hence show that the degree of cos(2π /n) is φ(n)/2. Use this to prove that cos(2π /7) and cos(π /9) has degree 3

(c) For which integer n is cos(2π /n) rational

(d) Use (b) to show that cos(2π /n) is of degree 3 only for n=7,9,14,18. Also find the n for which cos(2π /n) is of degree 4.

(e) Show that any isomorphisms of Q(cos(2π /n) are isomorphisms of Q(ζn).

(f) Show that for each k relatively prime to n there is an automorphism σk of Q(cos(2π /n) defined by σk(cos(2π /n))=cos(2kπ /n), and that every automorphisms of Q(cos(2π /n) is of this form.(compared with (e))

(g) How many of the automorphisms σk of Q(cos(2π /n) are actually distinct? Compare this number with the value of (Q(cos(2π /n) : Q) found in (b)

(h) Use (g) or otherwise, find fields Q(cos(2π /n) with automorphism groups of three and five elements.

Update:
(i) Consider the effect of restricting the automorphism of Q(ζn) to Q(cos(2π /n)) (compare with (f)) What is the kernel of the restriction map ?

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