Cyclic codes with lenght divisible by the field characteristic as invariant subspaces

In the theory of cyclic codes it is a common practice to require $(n,q)=1$, where $n$ is the word length and $F_q$ is the alphabet. However, much of the theory also goes through without this restriction on $n$ and $q$. We observe that the cyclic shift map is a linear operator in $F^n_q$. Our approach is to consider cyclic codes as invariant subspaces of $F^n_q$ with respect to this operator and thus obtain a description of cyclic codes in this more general setting.