Let K be the splitting field of x¹² - 1 over Q.
a. Find [K : Q].
b. Show that for σ ∈ G (K/Q), σ² is the identity automorphism. Classify G(K/Q) according to the Fundamental Theorem 11.12 of finitely generated abelian groups.

Data from Theorem 11.12

Every finitely generated abelian group G is isomorphic to a direct product of cyclic groups in the form

where the p_i are primes, not necessarily distinct, and the r; are positive integers. The direct product is unique except for possible rearrangement of the factors; that is, the number (Betti number of G) of factors Z is unique and the prime powers (p_i)^ri are unique.

Transcribed Image Text:

Z(p₁) X Z(p₂2)2 X.. + X Z(p) x ZxZx x 2,