number of monic irreducible polyynomials

**sidi** · May 25th 2009, 03:26 AM

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

**ThePerfectHacker** · May 25th 2009, 08:22 AM

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?

**sidi** · May 25th 2009, 09:23 AM

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

**NonCommAlg** · May 25th 2009, 10:50 AM

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.

**sidi** · May 25th 2009, 11:41 AM

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

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