# Test irreducibility of polynomial over finite field

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!

• 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

## 1 Answer

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 $\,\color{#c00}{x^5}\!-x-1 = \color{#c00}{\pm x} - x - 1\ne 0\,$ $\Rightarrow\!\Leftarrow$