Is there a general characterization of irreducible polynomials over a finite field?
I was going through a problem in finding whether $p(x):=x^7+x^5+1$ is irreducible over $\mathbb F_2[x]$ or not.
If the polynomial is of degree less than or equal to $3$ then we can easily find out if its irreducible or not by finding whether it has a root or not. In this case considering the polynomial $p(x)=f(x)\cdot g(x)$ we may be able to show the irreducibility but this doesn't seem to be a very great idea. Can anyone suggest a better idea ?