Different ways of proving a polynomial is irreducible in finite field. 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 ? 

Over a finite field $\mathbb F_p$, the product of all irreducible polynomials of degree $d$ is $x^{p^{d}}-x$, so this polynomial is irreducible iff it divides $x^{2^{7}}-x = x^{128}-x$. Now you can just check this by computation.
