I used this formula to calculate the number of irreducible polynomials for $\displaystyle n = 5$. Using it we conclude that there are 6 irreducible polynomials of this degree.
The problem I have though, is that when I tried calculating these irreducibles, I obtained the following 8 polynomials:
$\displaystyle x^5 + x + 1$
$\displaystyle x^5 + x^2 + 1$
$\displaystyle x^5 + x^3 + 1$
$\displaystyle x^5 + x^3 + x^2 + x + 1$
$\displaystyle x^5 + x^4 + 1$
$\displaystyle x^5 + x^4 + x^2 + x + 1$
$\displaystyle x^5 + x^4 + x^3 + x + 1$
$\displaystyle x^5 + x^4 + x^3 + x^2 + 1$
We know that if a degree 5 polynomial is reducible, then it must be divisible by one of $\displaystyle x,x+1,x^2 + x + 1$, yes? Taking the product of these yields the polynomial $\displaystyle x^4 + x$ over $\displaystyle Z_2$. Each of the above 8 polynomials $\displaystyle p(x)$ satisfies $\displaystyle gcd(x^4 + x,p(x)) = 1$, hence all 8 must be irreducible, right?
So whats going on
?