Wedderburn's theorem: "Every finite division ring is a field." This is really astonishing if you think of quaternions: nothing analogous in the finite case.
Then of course the classification of finite fields is also very beautiful: exactly one with p^n elements (p a prime and n an integer) and no others.
And as a bonus, Wedderburn's theorem is one of the crispest in all of mathematics: seven words ( or six and a half if you replace division ring by skew-field).
