I know that the AES MixColumn step is calculated as follows
$b(x) = (a(x)c(x)) \mod l(x)$
with $a(x)$ being the column to encode, $c(x)$ the fixed polynomial $c(x) = 3x^3 + x^2 + x + 2$ and $l(x) = x^4+1$ which is reducible over $GF(2^8)$ since $x^4 + 1 = (x+1)^4$.
I wondered about the purpose of the reducibility property of $l(x)$ and so I looked it up in my text book as well as on the internet but I could not find any further explanation on this. As it is my understanding only an irreducible polynomial $f(x)$ would be able to create a field $GF(2^8)[x]/(f(x))$. So my question is:
Is there a particular reason why l(x) must be reducible over $GF(2^8)$ in AES MixColumns?