Thread: determine irreducible quadratic polynomials

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

Originally Posted by mandy123

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

The polynomial factors into a product of monic irreducible polynomials of order dividing .
There are linear monic polynomials. Let be the number of monic irreducible quadradic polynomials.
Then by counting degrees of polynomials in
We see that .

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

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?

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 where (a power of prime). In this case the same theorem applies i.e. factors into monic irreducible polynomials having order dividing .