Linearized polynomial over a finite field ($L(X)={\sum }_{i=0}^d{a}_i{X}^{q^i}$)

Over a field containing ${\mathbb{F}}_q$, a sum of $q$-power monomials is ${\mathbb{F}}_q$-linear. If its leading coefficient and linear coefficient are nonzero, its root space has exactly $q^d$ elements and dimension $d$ over ${\mathbb{F}}_q$. Automorphisms of its splitting field act faithfully and linearly on that root space, embedding the Galois group in $GL(d,q)$.