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 $\displaystyle F_q[x]/(p(x))$ is a field if and only if $\displaystyle p$ is irreducible. However, the polynomial in question is reducible: $\displaystyle x^9+x^5+x^3+x+1=\left(1+x+x^2\right) \left(1+x^2+x^4+x^6+x^7\right)$, so $\displaystyle F_q[x] /(x^9+x^5+x^3+x+1)$ cannot be a field.