0

Find the number of monic square-free polynomials of degree j >=1 over the finite field GF(q) ?

I have no idea how to approach this. I was thinking if there was a way to write a monic polynomial uniquely, it might lead to something. Could someone help with this

Amrita
  • 860

1 Answers1

2

If monic polynomial $p(x)$ is not square-free, it can be written uniquely as $a(x) b(x)^2$ where $a(x)$ and $b(x)$ are monic, $a(x)$ is square-free and $b(x)$ has degree $\ge 1$. Considering the possible degrees of $b(x)$, you should be able to derive a recurrence, an equation for the number of square-free polynomials of degree $d$ in terms of the numbers with lower degrees.

You might also look up "squarefree polynomials" in the OEIS to get the answers for some values of $q$.

Robert Israel
  • 448,999
  • Could you give a more detailed explanation on why it could be written uniquely. Then how to approach. I am new to abstract algebra. – Amrita Sep 09 '16 at 16:05
  • 1
    Use the unique factorization into monic irreducibles.
    $a(x)$ is the product of the factors that occur to odd powers, $b(x)$ is the (monic) square root of $p(x)/a(x)$.     – Robert Israel Sep 09 '16 at 17:47