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An embedding of a subfield into an algebraically closed field extends over an algebraic extension: for a simple extension, map its generator to a root of the transported minimal polynomial, then iterate or use Zorn lemma. If the larger extension is normal over the fixed base, its image under the extended embedding is itself.

Over ${\mathbb{F}}_q$, the roots of the minimal polynomial of $\alpha \in {\mathbb{F}}_{q^n}$ are its distinct iterates $\alpha ,{\alpha }^q,{\alpha }^{q^2},\dots$ under the finite-field Frobenius automorphism.

A square matrix $A$ is invertible exactly when its minimal polynomial has nonzero constant term. If $m_A(t)=tq(t)$, then $Aq(A)=0$, and invertibility would contradict the minimality of $m_A$. Conversely, if $m_A(t)=tq(t)+c$ with $c\ne 0$, then $A[-q(A)/c]=I$.