Give a one-sentence synopsis of the proof of Theorem 19 .11. 

Data from Theorem 19.11  

Every finite integral domain is a field. 

Proof Let 

0, 1, a₁, ........a_n

be all the elements of a finite domain D. We need to show that for a ∈ D, where a ≠ 0, there exists b ∈ D such that ab = 1. Now consider 

a₁, aa₁. . ...., aa_n. 

We claim that all these elements of D are distinct, for aa_i = aa_j implies that a_i = a_j, by the cancellation laws that hold in an integral domain. Also, since D has no 0 divisors, none of these elements is 0. Hence by counting, we find that a₁, aa₁, • • •, aa_n are elements 1, a₁, •••,a_n in some order, so that either a1 = 1, that is, a = 1, or aa_i = 1 for some i. Thus a has a multiplicative inverse.