I just started doing some reading about multiplication in finite fields and i keep stumbling over one point: in the field G(2^8)
how does x^8 + x^4 + x^3 + x + 1 = 0 imply that x^8 = x^4 + x^3 + x + 1 ?
and, if addition is commutative, does this mean x^4 = x^8 + x^3 + x + 1 too?
As i said i am absolutely new to this topic, and i am wondering if this is some kind of special rule for polynomials or if i miss something basic i should know :/
thanks in advance for help! (dummy explanation please^^)
In a field of characteristic $2$, we have $1+1=0$, and therefore $a+a=0$ for all $a$. So $-a=a$. Nice, it is impossible to make a sign error.