# Finding a field extension in which every element has zero trace

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.

• As you've noted the only place to look is degree p (or a multiple) extensions in characteristic p. Looking up some results on purely inseparable extensions might help. – Alex J Best Feb 24 at 20:15

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.