Finding a field extension in which every element has zero trace

Question

Let F be a field extension of K, then F over K is a vector space, and for each a in F define f:F-->F as f(x)=ax, this is a linear transformation, define trace of a as trace of this linear transformation. If K has characteristic zero then clear 1 has trace n, which is non zero. Are there fields F and K as above, such that every element of F has zero trace.

Answer 1

Let $F=\Bbb{F}_p(x)$ and $K=\Bbb{F}_p(x^p)$. A basis of the extension $F/K$ consists of $1,x,\ldots,x^{p-1}$. The element $1$ has $tr^F_K(1)=0$ because the $p\times p$ identity matrix has trace zero. The element $x^i, i=1,2,\ldots,p-1$, has trace zero because its minimal polynomial $m_i(T)$ over $K$ is $$(T-x^i)^p=T^p-(x^p)^i\in K[T],$$ and the coefficient of the degree $p-1$ term is manifestly zero.

The trace $tr^F_K$ is $K$-linear, so if it vanishes on a basis it vanishes everywhere.