  In this paper we discuss the permutational property of polynomials of the
form $f(L(x))+k(L(x))\cdot M(x)\in \mathbb F_{q^n}[x]$ over the finite field
$\mathbb F_{q^n}$, where $L, M\in \mathbb F_q[x]$ are $q$-linearized
polynomials. The restriction $L, M\in \mathbb F_q[x]$ implies a nice
correspondence between the pair $(L, M)$ and the pair $(g, h)$ of conventional
$q$-associates over $\mathbb F_q$ of degree at most $n-1$. In particular, by
using the AGW criterion, permutational properties of our class of polynomials
translates to some arithmetic properties of polynomials over $\mathbb F_q$,
like coprimality. This relates the problem of constructing PPs of $\mathbb
F_{q^n}$ to the problem of factorizing $x^n-1$ in $\mathbb F_q[x]$. We then
specialize to the case where $L(x)$ is the trace polynomial from $\mathbb
F_{q^n}$ over $\mathbb F_q$, providing results on the construction of
permutation and complete permutation polynomials, and their inverses. We
further demonstrate that the latter can be extended to more general linearized
polynomials of degree $q^{n-1}$.
