How to construct a cubic monic irreducible polynomial over $\mathbb{F}_p[x]$? $p$ is prime.

Question

I'd like to know that how to construct a cubic monic irreducible polynomial over $\mathbb{F}_p[x], p$ is a prime number.

It is known that the polynomial is irreducible iff it has no roots in $\mathbb{F}_p$. Suppose we have the following cubic polynomial, $x^3-x+a, a\in \mathbb{F}_p$. How should we determine the value of $a$ ?

Could you give a few examples? If there is a general solution, it is the best.

Thanks for your answer.

Answer 1

One method is by trial and error. Write down a monic cubic polynomial which has no zero in $\Bbb Z_p$. This polynomial is already irreducible. If not, it must factor into a polynomial of degree 1 and one polynomial of degree 2 (which may or may not be irreducible). The polynomial of degree 1 has a zero of the polynomial of degree 3.

Answer 2

For small values of $p$, it's usually not too bad to write out a complete list of monic irreducible linear polynomials and monic irreducible quadratic polynomials in $\mathbb{F}_p[x]$. Then, to get a monic cubic irreducible polynomial in $\mathbb{F}_p[x]$ it suffices to check via trial and error with new coefficients whether any existing monic irreducible linear or monic irreducible quadratic polynomials divides the new monic cubic polynomial you've written down.