Squaring is linear in Galois Field $2$

Question

In the context of cyclic codes and BCH, consider the generation function of a codeword $c = [c_0 \ldots c_n-1]$ to be $c(x) = c_0 + \cdots + c_{n-1}x^{n-1}$.

Now, in the finite field $F_2[x]$, why is $c(\alpha^2)=c(\alpha)^2, c(\alpha^4)=c(\alpha)^4$ but $c(\alpha^3) \neq c(\alpha)^3$ (with $\alpha$ being primitive element of $F_2[x]/x^3+x+1$)?

I also came across the statement "In fields of characteristic $2$, squaring is a linear function". Why is this true?

Thanks.

Answer 1

because $(a+b)^2=a^2+2ab+b^2=a^2+b^2$ because the field has characteristic 2.