I am reading a text, where it says that $X^5-X-1$ is irreducible modulo 3. I am not sure how I can know that. Could someone help?

By the way, is there some practical trick to judge whether a polynomial irreducible over a finite field in general? Because when I calculate the galois group I find this information is very important, so I am very curious. Thanks!

  • 2
    $\begingroup$ You can do it the crude way. There are no roots, so without loss of generality we can look for a decomposition into a monic cubic times a monic quadratic. Then see what the coefficients should be. Somewhat painful, but not lengthy. $\endgroup$ – André Nicolas Jul 30 '16 at 16:47
  • $\begingroup$ @AndréNicolas Ops, sure, thanks! But I am still curious about other finite fields. $\endgroup$ – user330928 Jul 30 '16 at 16:48
  • $\begingroup$ There is theory that (sometimes) helps. But at this stage the best one can do is probably clever manipulation from basics. $\endgroup$ – André Nicolas Jul 30 '16 at 16:57

Hint $ $ It has no roots so no linear factors, so if it splits it has an irreducible quadratic factor $g$. In $\,\Bbb F_9 = \Bbb F_3[x]/g\,$ we have $\,x^8 = 1\,$ so $\,\color{#c00}{x^4 = \pm1}\,$ so $\,\color{#c00}{x^5}\!-x-1 = \color{#c00}{\pm x} - x - 1\ne 0\,$ $\Rightarrow\!\Leftarrow$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.