Give a synopsis of the proof of Corollary 23.6. 

Data from 23.6 Corollary 

If G is a finite subgroup of the multiplicative group (F^*, ·) of a field F, then G is cyclic. In particular, the multiplicative group of all nonzero elements of a finite field is cyclic. 

Proof: By Theorem 11.12 as a finite abelian group, G is isomorphic to a direct product Z_d1 x Z_d2 x • • • x Z_dr, where each d_i is a power of a prime. Let us think of each of the Z_di as a cyclic group of order d_i in multiplicative notation. Let m be the least common multiple of all the d_i for i = 1, 2, · · ·, r; note that m ≤ d₁d₂ · · ·d_r. If a_i ∈ Z_di then a_i^di = 1, so a_i^m = 1 since d_i divides m. Thus for all a ∈ G, we have α^m = 1, so every element of G is zero of x^m - 1. But G has d₁d₂ • • • d_r elements, while x^m - 1 can have at most m zeros in the field F by Corollary 23.5, so m ≥ d₁d₂ · · · d_r. Hence m = d₁d₂ · · · d_r, so the primes involved in the prime powers d₁ , d₂, · · •, d_r are distinct, and the group G is isomorphic to the cyclic group Z_m.