Give a synopsis of the proof of Corollary 23.5.

Data from 23.5 Corollary 

A nonzero polynomial f(x) ∈ F[x] of degree n can have at most n zeros in a field F.  

Proof The preceding corollary shows that if a₁ ∈ F is a zero off (x ), then f(x) = (x - a₁)q₁(x), where, of course, the degree of q₁ (x) is n - 1. A zero a₂ ∈ F of q₁ (x) then results in a factorization f(x) = (x - a₁) (x - a₂) q₂(x)

Continuing this process, we arrive at f(x) = (x - a₁) · · · (x - a_r)q_r(X), where q_r(x) has no further zeros in F. Since the degree of f(x) is n, at most n factors (x - a_i) can appear on the right-hand side of the preceding equation, so r ≤ S n. Also, if b ≠ a_i for i = 1, ···,rand b ∈ F, then f(b) = (b - a₁) · · · (b - a_r)q_r(b) ≠ 0,  since F has no divisors of 0 and none of b - a_i or q_r(b) are 0 by construction. Hence the a_i for i = 1, • • •, r ≤ n are all the zeros in F of f(x).