Is there a formula for quickly calculating the number of irreducible polynomials of degree over ?
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 ?