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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!

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marked as duplicate by Jyrki Lahtonen abstract-algebra Oct 6 '18 at 14:58

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    $\begingroup$ I wrote an article about this that might be useful or at least interesting. $\endgroup$ – MJD Oct 6 '18 at 15:06