It is basic that the norm map $N:\mathbf{F}\_{q^n}^\* \to \mathbf{F}\_q^\*$ is surjective for finite fields. In fact $N(x) = x^{(q^n-1)/(q-1)}$. How well does this simple fact extend to subspaces?


A basic example is an intermediate extension $\mathbf{F}\_{q^d}$. On $\mathbf{F}\_{q^d}^\*$ we have
$$N(x) = \left(x^{(q^d-1)/(q-1)}\right)^{(q^n-1)/(q^d-1)} = \left(x^{(q^d-1)/(q-1)}\right)^{n/d}$$
since the term in the brackets is in $\mathbf{F}\_q^\*$ and $(q^n-1)/(q^d-1) \equiv n/d \pmod {q-1}$. So $N$ is surjective on $\mathbf{F}\_{q^d}^\*$ if and only if $(n/d, q-1) = 1$. In particular $N$ fails to be surjective on a subspace of dimension $n/2$ whenever $n$ is even and $(n/2, q-1) > 1$.


As a sort of converse note that if $(n,q-1)=1$ then $N$ is surjective on every one-dimensional subspace.



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> Is it true that if $V \leq \mathbf{F}\_{q^n}$ is a $\mathbf{F}\_q$-rational subspace of dimension $>n/2$ then $N$ is surjective on $V$?
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Equivalently, if $\dim\_{\mathbf{F}\_q} V > n/2$, can we always find $x^{q-1} \in V$?



Surjectivity of norm map on subspaces of finite fields
The number of elements with norm $a$ is $$\frac{1}{q-1} \sum\_{\chi: \mathbb F\_q^\times \to \mathbb C^\times} \sum\_{x \in V} \chi(Nx) \overline{\chi(a)}$$ The summand vanishes unless $\chi$ has order dividing $n$ so there are at most $gcd(n,q-1)$ terms. One of the terms has size $q^{\dim V}$ so it is sufficient that the other terms have size $<q^{ \dim V} / (\gcd (n,q-1)-1)$.


Each term is an average of $q^{n-\dim V}$ different Gauss sums associated to the additive characters that vanish on $V$ and hence has size $\leq q^{n/2}$.


So it suffices that $q^{ \dim V -n/2} > \gcd(n,q-1)-1$. This is automatically satisfied for $\dim V \geq n/2+1$ and is satisfied for $\dim V= (n+1)/2$ as long as $q$ is sufficiently large with respect to $n$.


This method is the same as Felipe Voloch's answer, because the eigenvalues in the curve he writes down are Gauss sums, except that we get a savings by dropping the unnecessary multiplicative characters.


It is likely possible to improve the dependence on $q$ somewhat by using etale cohomology theorem (the key thing is not to view this as a $1$-dimensional variety over $\mathbb F\_{q^n}$ but as an $n$-dimensional variety over $\mathbb F\_q$). But this will probably lead to some loss in $n$ so I don't know whether tit will improve the $\dim (n+1)/2$ case.


