# 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!

## marked as duplicate by Jyrki Lahtonen abstract-algebra StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Oct 6 '18 at 14:58

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