  Let $q$ be a prime power and $\mathbb F_{q^n}$ be the finite field with $q^n$
elements, where $n>1$. We introduce the class of the linearized polynomials
$L(x)$ over $\mathbb F_{q^n}$ such that
$$L^{(t)}(x):=\underbrace{L(L(\cdots(x)\cdots))}_{t \quad\text{times}}\equiv
0\pmod {x^{q^n}-x}$$ for some $t\ge 2$, called nilpotent linearized polynomials
(NLP's). We discuss the existence and construction of NLP's and, as an
application, we show how to construct permutations of $\mathbb F_{q^n}$ from
these polynomials. For some of those permutations, we can explicitly give the
compositional inverse map and the cycle structure. This paper also contains a
method for constructing involutions over binary fields with no fixed points,
which are useful in block ciphers.
