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
Answer
This actually brings up a subtle point. What do we mean by $5$ in a finite field? Or if you choose to define $5$ in terms of $1 ~(5=1+1+1+1+1)$, then what do we mean by $1$?
One answer is to define $5$ in terms of equivalence classes. Say that two integers $m$ and $n$ are equivalent if $p \vert (m-n).$ First, you prove this really is an equivalence relation on the integers. Then you define $[m]+[n]=[m+n]$ and $[m][n]= [mn]$. So by $5$ we actually mean the equivalence class $[5]$.
You need to prove that your field operations are well-defined (you get the same answer no matter which representative of an equivalence class you choose) and that $[0]$ and $[1]$ really are the additive and multiplicative identities, as you'd expect. But once you've done that, you can see that $[4]+[1]=[5]$ (and usually we abuse notation by dropping the brackets) because we've defined it that way.
No comments:
Post a Comment