# Thread: For p = 3, find the number of monic irreducible polynomials in...?

1. ## For p = 3, find the number of monic irreducible polynomials in...?

1.) For p = 3, find the number of monic irreducible polynomials in Zp[x] of the following degrees

(a) 5
(b) 9
(c) 10

2.) Repeat Exercise 1 for a general prime p. Your answers will be formulas in p.

Can anybody help me out, at least with part 1!? There must be any easier way than writing them all out.

2. ## Re: For p = 3, find the number of monic irreducible polynomials in...?

I have this formula from my problem session
$\displaystyle (p-1)p^k$ where $\displaystyle k:=deg(p(x))$

so there are
$\displaystyle (p-1)p^k=(3-1)3^5=486$ monic polynomials in $\displaystyle \mathbb{Z}_{3}$of $\displaystyle deg 5$

$\displaystyle (3-1)3^9=39366$ monic polynomials in $\displaystyle \mathbb{Z}_{3}$of $\displaystyle deg 9$

$\displaystyle (3-1)3^{10}=118098$ monic polynomials in $\displaystyle \mathbb{Z}_{3}$of $\displaystyle deg 10$

so it would take quite long time to list them all by hand..