The theorem statement is: a finite field of characteristic $p$ has $p^n$ elements. I found this very simple proof in the book "Ling, San; Xing, Chaoping; Coding Theory - A First Course". But I don't understand the highlighted step. Doesn't that require $(b_2-a_2)|(a_1-b_1)$?
Hint: The field is a vector space of finite dimension over its prime sub-field.