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

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.

• There is always an irreducible polynomial of the form $x^3-x-a$. That's because $x^3-x$ has the three obvious zeros, so it cannot give rise to a surjective function. So if $a$ is not in the range, the polynomial has no zeros and, being cubic, is irreducible. Commented Apr 27, 2022 at 21:28
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.
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.