# Thread: number of monic irreducible polyynomials

1. ## number of monic irreducible polyynomials

I need help with irreducible polynomials. Could anybody help me please?
The problem is:
Find the pattern for the number of monic irreducible polynomials of degree 6 in $Z_2[x]$!
Is it different to look for monic irreducible and irreducible polynomials?
Thank you very much

2. Originally Posted by sidi
I need help with irreducible polynomials. Could anybody help me please?
The problem is:
Find the pattern for the number of monic irreducible polynomials of degree 6 in $Z_2[x]$!
Is it different to look for monic irreducible and irreducible polynomials?
Thank you very much
In the case of $\mathbb{Z}_2[x]$ all polynomials are monic. Therefore, finding the monic irreducible polynomials is equivalent to finding the irreducible polynomials. There is a formula by Gauss that gives you the number of such irreducible polynomials. Are you asking for a formula?

3. yes, I think I am looking for that formula...

4. Originally Posted by sidi

yes, I think I am looking for that formula...
$\frac{1}{6} \sum_{d \mid 6} \mu(6/d) 2^d = 9,$ where $\mu$ is the Mobius function.

5. Is there any other way how to derive the number of these polynomials without using this formula?