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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!

user330928
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    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. – André Nicolas Jul 30 '16 at 16:47
  • @AndréNicolas Ops, sure, thanks! But I am still curious about other finite fields. – user330928 Jul 30 '16 at 16:48
  • There is theory that (sometimes) helps. But at this stage the best one can do is probably clever manipulation from basics. – André Nicolas Jul 30 '16 at 16:57

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Hint $\:\!\color{#0a0}f\:\!$ has no roots $\in\Bbb F_3$ so no linear factors. If it splits it has an irreducible $\rm\color{#90f}{quadratic}$ factor $\color{#90f}g.\,$ In $\,\Bbb F_9\! = \Bbb F_3[x]/\color{#90f}g\!:\: $ $\underbrace{x^8\! = 1}_{\!\!\!\!\!\!\!\!\!\large \Rightarrow\,\ \color{#c00}{{x^4}\ =\ \pm1}}\,$ so $\,0\! =\underbrace{\color{#c00}{x^4}\color{0a0a}x\!-\!x\!-\!1}_{\large \color{#0a0}{\deg f\ \le\ 1}}\, $ so $\,\color{#90f}g\mid\color{#0a0} f\,$ in $\,\Bbb F_3[x],\,$ contra $\,\color{#90f}{\deg g \!=\! 2},\,\color{#0a0}{\deg f\!\le\! 1}$.

Remark $ $ We used $ f^{\large \color{#c00}8}\! = 1\,$ for $\,0\!\neq \!f\in \Bbb F_9,\,$ an analogue of Fermat's little Theorem, which is true by $\,\Bbb F_9$ has multiplicative group $\,\Bbb F_9^*$ of size $\,\color{#c00}8 = 9\color{#0a0}{-1}\,$ ($\rm non\color{#0a0}{zero}$ elements are invertible in a field), so Lagrange's Theorem $\Rightarrow f^{\large \color{#c00}8}\! = 1\,$ for all $\,f\!\neq\! 0.$

Above is essentially a special case of a general polynomial irreducibility test over finite fields - which is an an efficient analog of the impractical Pocklington-Lehmer integer primality test.

Bill Dubuque
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