 Linearized polynomial over a finite field

Over a field $K$ containing ${\mathbb{F}}_q$, a q-linearized polynomial isThe Frobenius endomorphism gives $L(u+v)=L(u)+L(v)$ and $L(cu)=cL(u)$ for $c\in {\mathbb{F}}_q$, so its polynomial function is ${\mathbb{F}}_q$-linear. If $a_0\ne 0$, its formal derivative is the nonzero constant $a_0$. Thus it has exactly $q^d$ distinct roots of a polynomial in an algebraic closure, forming a vector space of dimension $d$ over ${\mathbb{F}}_q$.

The chosen coefficient field need not contain all those roots. For instance, $X^{q^2}-X$ has $q^2$ roots in ${\mathbb{F}}_{q^2}$ but only $q$ in ${\mathbb{F}}_q$. The Galois group of the splitting field over $K$ acts faithfully and linearly on the full root vector space, embedding in $GL(d,q)$. Faithfulness holds because the root vectors generate the splitting field.