I used this formula to calculate the number of irreducible polynomials for
. 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:
We know that if a degree 5 polynomial is reducible, then it must be divisible by one of
, yes? Taking the product of these yields the polynomial
over
. Each of the above 8 polynomials
satisfies
, hence all 8 must be irreducible, right?
So whats going on
?