ONB1 identity element

Rosing wrote on page 144:

In an optimal normal basis, adding 1 is the same as taking a complement

And finally figuring this out after finding out that the identity element of ONB1 is $x^{2^{n-1}}+...+x^{2^{1}}+x^{2^{0}}$. If we have n = 4, the identity element or “1” is 1111 ($x^{8}+x^{4}+x^{2}+x$).

That’s why if we have 0011 in normal basis ($x^2+x$), adding it with “1” so it becomes 0011 + 1111 = ($(x^2+x) + (x^8+x^4+x^2+x)$ = $x^8+x^4$ = 1100.

Hoooraaaay!