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Can someone briefly explain Field theory to me?

Ie, in this book, it says that 1+1 = 0 when field : F (subscript 2)

And you can create a finite field for any prime number p. I don't understand this lol.

I also got this table:

+ 0 1 2 3

00 1 2 3

11 0 3 2

22 3 0 1

33 2 1 0

and Multiplication:

x 0 1 2 3

00 0 0 0

10 1 2 3

20 2 3 1

30 3 1 2

F(Subscript 4)

How do I go about finding these numbers? 3x3 is 9 as far as Im concerned, unless he is using clock arithmetic, in which case 3x3 would be 1 on a 4 Clock... its 2 here T___T

Help appreciated :D

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# Need help on Field theory

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