help needed , I have found this one in a book but it has no answer.
how can I show for any finite field F_q of even characterestic , the ring F_q[x] /(x^9+x^5+x^3+x+1) cannot be a field?
Just to clarify: the characteristic of a field is either zero or a prime number, so "even characteristic" really just means characteristic 2.
Recall that is a field if and only if is irreducible. However, the polynomial in question is reducible: , so cannot be a field.