Referring to Example 6, show that the automorphism f has
Chapter 32, Problem 9E(choose chapter or problem)
Referring to Example 6, show that the automorphism f has order 6. Show that ? + ? -1 is fixed by ?3 and ?3 + ?5 +?6 is fixed by ?2.(This exercise is referred to in this chapter.)Reference Example 6Let ? = cos(2?/7) 1 i sin(2 ? /7), so that ?7 = 1, and consider the field Q(?). How many subfields does it have and what are they? First, observe that Q(?) is the splitting field of x7 - 1 over Q, so that we may apply the Fundamental Theorem of Galois Theory. A simple calculation shows that the automorphism f that sends ? to ?3 has order 6. Thus,
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