(1) In $\mathbb{F}_{2^n}$ with odd n it should be shown that half of the Trace values are 0 and the other part is 1 with help of the additivity of the Trace. (2) Now n shall be even. Now the distribution of the Trace values of the elements is the same as (1). For the proof it should be used that if n is even $\mathbb{F}_{2^n}$ contains a sub field which is isomorph to n $\mathbb{F}_{2^2}$
Has anybody an idea how to start ?