Finding the order of an irreducible polynomial $f$ in $F_3[x]$ of degree 4?

Question

The technique I am using is based on the long division of $x^e - 1$ (e is to be the order) which is really tiresome. So what the other methods (efficient)?

Answer 1

The roots of your irreducible quartic polynomial $f(x)$ reside in the field $\Bbb{F}_{81}$. The multiplicative group of that field is cyclic of order 80. Therefore the order of your polynomial is a factor of $80$. Because your polynomial does not split into a product quadratics and lower, its order is not a factor of $8$. That leaves $10,16,20,40$ and $80$ as alternatives. If you can check that the order is not a factor of $40$ or $16$, then you can already conclude that it has to be $80$.

But don't do long division. You can reasonably quickly calculate the residue of $x^e$ modulo $f(x)$ by the good ole square-and-multiply algorithm.