Show that there exists one and only one field with three elements.

If I had to guess, it would be the cyclic subgroup

. Am I anywhere near on target? I figure this since each element would have inverses in the field. I am aware that

whereby p is prime is a field of p elements. Obviously, 3 is prime. Should I solve this generally for p and then simply state that 3 is an instance of a prime number? There has to be something I am either forgetting or simply not getting. The part I am confused about is HOW to shoe that there is ONE and ONLY ONE field with 3 elements. Thanks.