I did some search over the website and in google, but couldn't find the answer, even though I'm sure it exists, so maybe I did not look for the right key words.
Anyway, the question is how do I prove that in a finite field - GF(p), every exponent of a sum, is the sum of exponents?
Or, in a formula: $(a+b)^p=a^p+b^p$
I know it's proven with the binomial theorem, but I'm not sure why for example: $\binom{p}{0} $ is 1 (I do understand why it's 1 in a non-finite field). The way I see it is:
$\binom{p}{0}=\frac{p!}{(p-0)!0!}=\frac{p\cdot(p-1)!}{p!\cdot1}=\frac{0\cdot (p-1)!}{0\cdot(p-1)!}=\frac{0}{0}$
Now it seems that the result is not defined (or is it? in a finite field...).. while in a non finite field it goes like this:
$\binom{p}{0}=\frac{p!}{(p-0)!0!}=\frac{p!}{p!\cdot1}=\frac{1}{1}=1$
I would appreciate your help on the rest of the proof as well.. I know there is some main idea here..I'm new to finite fields, so maybe I'm missing something here...