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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.

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  • $\begingroup$ The first answer of math.stackexchange.com/questions/2325588/… should help $\endgroup$
    – Bailey
    Commented Apr 27, 2022 at 12:07
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    $\begingroup$ Welcome! People are likely to answer if you show some thought of how you might attempt. Even mention some examples/ properties of reducible monic cubics. $\endgroup$
    – Jalog_the_Martian
    Commented Apr 27, 2022 at 12:07
  • $\begingroup$ See this paper and also mathoverflow.net/questions/144974/… $\endgroup$
    – lhf
    Commented Apr 27, 2022 at 12:55
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    $\begingroup$ 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. $\endgroup$
    – Jyrki Lahtonen
    Commented Apr 27, 2022 at 21:28

2 Answers 2

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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.

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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.

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