I have just begun studying finite fields today, and it is clear in GF(2) why 1+1=0. (I just show that 1+1 can't equal 1, or 1=0, which contradicts an axiom that states that 1 is not 0).
If we interpreted these symbols "1", "+", "1", "0" as we would in primary school, clearly this breaks arithmetic rules in Real numbers.
Given that, I have lost all confidence in how arithmetic can be applied in a finite field. How do I even know how to do basic arithmetic on GF(n) where n is prime? For example, for GF(7), how do I even know that 4+1=5? Can anyone show with just the 9 axioms of finite fields that 4+1=5?
Axioms: associativity of addition, additive identity, additive inverse, commutatitivity of addition, associativity of multiplication, multiplicative inverse, commutatitivity of mulitplication, distributive law