# Finding all degree 4 irreducible polynomials over $\mathbb{Z_2}$ [duplicate]

So I'm looking for all the irreducible degree four polynomials over $$\mathbb{Z_2}$$. Now, there are only 16 total degree four polynomials over $$\mathbb Z_2$$, and so I could go through and check each one, first checking to see if it has a root, and if it does than it must not be irreducible. After I narrow my collection down to the ones without roots, I could try to factor each as a quadratic by setting it equal to an arbitrary product of quadratics and then matching coefficients. Is this the best way to do this or is there an insight that I am missing that could speed up this process? Thanks!