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 ? 

The given polynomial is in fact not irreducible. There is at least one decomposition:
$$
(x^2+x+1)\cdot(x^5+x^4+x^3+x+1) = x^7+x^5+1
$$
This can be found by resolving the equality for coefficients:
$$
(x^2+ax+1)\cdot(x^5+bx^4+cx^3+dx^2+ex+1) = x^7+x^5+1
$$
which, equating term by term, and ignoring the terms of degree $0$ and $7$:
$$
(a+b)x^6 + (1+ab+c)x^5 + (b+ac+d)x^4 + (c+ad+e)x^3 + (d+ae+1)x^2 + (e+a)x = x^5
$$
so
$$
a+b=0 \\
ab+c=1 \\
b+ac+d = 0\\
c+ad+e=0\\
d+ae=1\\
e+a=0
$$
which are more than enough to find the solution, given that in $\mathbb{Z}_2$, $a^2=a$ and $a+a=0$ for any $a$.
