If $K$ has characteristic $p>0$, an
[Artin-Schreier extension][1]
$L/K$ is cyclic of degree $p$, and not of the indicated form. Such
an $L$  has the form $K(a)$ for a root $a$ of a polynomial of the 
form $X^p - X - \alpha$ for a suitable $\alpha \in K$; the polynomial
then splits over $L$ since its roots are all $a+i$ for $i \in \mathbf{F}_p$
(in this way you can see the action of the Galois group $\mathbf{Z}/p$ on the roots).

An example of such an extension is $\mathbf{F}_4/\mathbf{F}_2$ since $\mathbf{F}_4 = \mathbf{F}_2(a)$ for a root $a$ of $X^2 - X - 1 = X^2 + X +1$.


  [1]: http://en.wikipedia.org/wiki/Artin%25E2%2580%2593Schreier_theory