Repeat Exercise 4 for a primitive 7th root of unity in C. Data from exercise 4

Repeat Exercise 4 for ζ a primitive 7th root of unity in C.

Data from exercise 4

Let ζ be a primitive 5th root of unity in C.

a. Show that Q(ζ) is the splitting field of x⁵ - 1 over Q.

b. Show that every automorphism of K = Q(ζ) maps ζ onto some power ζ^r of ζ. 

c. Using part (b), describe the elements of G(K/Q).

d. Give the group and field diagrams for Q(ζ) over Q, computing the intermediate fields as we did in Examples 54.3 and 54.7.