Noncommutative polynomial maps. (Q2909806)

Let \(p\) be a prime and \([i]=1+p+\cdots+p^{i-1}\). Suppose that \(q=p^n\) and \(\mathbb F_q\) is the finite field with \(q\) elements. Denote by \(\theta\) its Frobenius automorphism. The map \(\theta\) can be extended to the algebra of polynomials \(\mathbb F_q[t]\). Denote by \(\mathbb F_q[x^{[\;]}]\) the set of all polynomials \(\sum_{i\geqslant 0}\alpha_ix^{[i]}\). Evaluating each polynomial \(f\in\mathbb F_q[t;\theta]\) at \(x\) we obtain a polynomial \(f^{[\;]}(x)\in\mathbb F_q[x]\), namely \(f(t)(x)=f^{[\;]}(x)\). It is shown that a polynomial \(f(t)\in\mathbb F_q[t]\) is irreducible if and only if the polynomial \(f^{[\;]}(x)\in\mathbb F_q[x^{[\;]}]\subset\mathbb F_q[x]\) has no nontrivial factors from \(\mathbb F_q[x^{[\;]}]\).