1. ## determine irreducible quadratic polynomials

Let p be a prime. Determine the number of irreducible quadratic polynomials over Z_p

2. Originally Posted by mandy123
Let p be a prime. Determine the number of irreducible quadratic polynomials over Z_p
The polynomial $x^{p^2} - x$ factors into a product of monic irreducible polynomials of order dividing $2$.
There are $p$ linear monic polynomials. Let $n$ be the number of monic irreducible quadradic polynomials.
Then by counting degrees of polynomials in $x^{p^2} - x = \prod_{\deg p(x) | 2}p(x)$
We see that $p + 2N = p^2 \implies N = \tfrac{1}{2}(p^2 - p)$.

(This formula can be generalized to polynomials of degree $m$ by applying Mobius inversion formula)

3. So then what if p is not a prime, what if we are told to
Determine the number of irreducible quadratic polynomials over Z_p?
how would that change the answer?

4. Originally Posted by mandy123
So then what if p is not a prime, what if we are told to
Determine the number of irreducible quadratic polynomials over Z_p?
how would that change the answer?
Finite fields have orders of power of primes. Thus, your question would be to count the number of irreducible quadradics over $\mathbb{F}_q$ where $q=p^n$ (a power of prime). In this case the same theorem applies i.e. $x^{q^2} - x$ factors into monic irreducible polynomials having order dividing $2$.