= Quadratic constant extension of a q-squared linearized splitting field
{title2=$\mathrm{Gal}(E/\mathbb F_{q^2}(T))\triangleleft\mathrm{Gal}(E/\mathbb F_q(T))$}
For even $d=2n$, the roots of $X^{q^d}+X^{q^2}+TX$ form an $n$-dimensional <vector space> over $\mathbb F_{q^2}$. Ratios $(c\alpha)/\alpha$ for a nonzero root $\alpha$ show that the <splitting field> contains every $c\in\mathbb F_{q^2}$. The subgroup fixing that quadratic constant extension is normal of index two and lies in $GL(n,q^2)$; applying projective transitivity over $\mathbb F_{q^2}(T)$ makes it contain $SL(n,q^2)$. It therefore normalizes that <special linear group>, while the full group acts semilinearly.
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